Analytical modeling and correction of steady state relative permeability experiments with capillary end effects – An improved intercept method, scaling and general capillary numbers

Steady state relative permeability experiments are performed by co-injection of two fluids through core plug samples. Effective relative permeabilities can be calculated from the stabilized pressure drop using Darcy’s law and linked to the corresponding average saturation of the core. These estimated relative permeability points will be accurate only if capillary end effects and transient effects are negligible. This work presents general analytical solutions for calculation of spatial saturation and pressure gradient profiles, average saturation, pressure drop and relative permeabilities for a core at steady state when capillary end effects are significant. We derive an intuitive and general “intercept” method for correcting steady state relative permeability measurements for capillary end effects: plotting average saturation and inverse effective relative permeability (of each phase) against inverse total rate will give linear trends at high total rates and result in corrected relative permeability points when extrapolated to zero inverse total rate (infinite rate). We derive a formal proof and generalization of the method proposed by Gupta and Maloney (2016) [SPE Reserv. Eval. Eng. 19, 02, 316–330], also extending the information obtained from the analysis, especially allowing to calculate capillary pressure. It is shown how the slopes of the lines are related to the saturation functions allowing to scale all test data for all conditions to the same straight lines. Two dimensionless numbers are obtained that directly express how much the average saturation is changed and the effective relative permeabilities are reduced compared to values unaffected by end effects. The numbers thus quantitatively and intuitively express the influence of end effects. A third dimensionless number is derived providing a universal criterion for when the intercept method is valid, directly stating that the end effect profile has reached the inlet. All the dimensionless numbers contain a part depending only on saturation functions, injected flow fraction and viscosity ratio and a second part containing constant known fluid, rock and system parameters such as core length, porosity, interfacial tension, total rate, etc. The former parameters determine the saturation range and shape of the saturation profile, while the latter number determines how much the profile is compressed towards the outlet. End effects cause the saturation profile and average saturation to shift towards the saturation where capillary pressure is zero and the effective relative permeabilities to be reduced compared to the true relative permeabilities. This shift is greater at low total rate and gives a false impression of rate-dependent relative permeabilities. The method is demonstrated with multiple examples. Methodologies for deriving relative permeability and capillary pressure systematically and consistently, even based on combining data from tests with different fluid and core properties, are presented and demonstrated on two datasets from the literature. The findings of this work are relevant to accurately estimate relative permeabilities in steady state experiments, relative permeability end points and critical saturations during flooding or the impact of injection chemicals on mobilizing residual phase.publishedVersio

Early and Late Time Analytical Solutions for Co-current Spontaneous Imbibition and Generalized Scaling

We propose an explicit analytical solution for 1D cocurrent (COC) spontaneous imbibition (SI) in which a core is exposed to water (inlet) and oil (outlet). The system is described using an advection-capillary diffusion transport equation combined with a pressure equation. By ignoring the capillary diffusion term in the transport equation, the analytical solution follows in terms of Buckley-Leverett (BL) saturation profiles. The capillary force appears in the pressure equation and determines the advective term of the transport equation. The time when the front reaches the outlet (critical time) is calculated and used for scaling. The solution is extended to after critical time (late time) by maintaining the BL profile inside the system, thus preserving continuity in recovery and spatial profiles. The solution is characterized by an effective total mobility and capillary pressure (incorporating the entire saturation functions), both constant at early time (before critical time). At late times, they change dynamically. The model states that the imbibition rate can increase, decrease, and stay constant with time based on a new mobility ratio being less than, more than, or equal to unity, respectively. The ratio also indicates effectiveness of oil displacement. The square root of time recovery is a special case only seen for a (very) favorable mobility ratio. The model predicts that COC imbibition scales with the square of length both at early and late times and that the solution can scale saturation functions. The analytical solution was compared against numerical simulations of the full system. The new mobility ratio reflected the evolution in COC recovery better than total recovery. The analytical solution showed a too-high imbibition rate at a favorable mobility ratio. The diffusion term is important then due to strong saturation gradients, and the resulting smoothened profile yields a lower imbibition rate from the pressure equation. The analytical solution showed a too-low imbibition rate at early times for unfavorable mobility ratio due to not accounting for rapid early countercurrent (COUC) production. The analytical solution predicted a too-high imbibition rate at late times because the BL profile does not capture the oil mobility restriction at the outlet at late times. The time of water reaching the outlet was underestimated by a factor ∼ 2 for strongly water-wet (SWW) simulations and ∼ 10 for mixed-wet (MW) simulations. Scaling recovery with length squared was exact for all times. Scaling recovery until water reaching the outlet demonstrated consistency across saturation functions and viscosities. The analytical solution could match literature experimental data and produce corresponding saturation functions. To our knowledge, previous analytical solutions have only considered infinite-acting systems (early time), assumed piston-like displacement (PLD) (uniform saturations on both sides of a saturation shock front) or are implicit, thus not providing more insight than numerical simulations.acceptedVersio

Steady State Gas Flow from Tight Shale Matrix Subject to Water Blocking

This work studies 1D steady-state flow of gas from compressible shale matrix subject to water blocking toward a neighboring fracture. Water blocking is a capillary end effect causing wetting phase (e.g., water) to accumulate near the transition from matrix to fracture. Hydraulic fracturing is essential for economical shale gas production. Water is frequently used as fracturing fluid, but its accumulation in the matrix can reduce gas mobility and production rate. Gas transport is considered at a defined pressure drop. The model accounts for apparent permeability (slip), compressibility of gas and shale, permeability reduction, saturation tortuosity (reduced relative permeability upon compaction), and multiphase flow parameters like relative permeability and capillary pressure, which depend on wettability. The behavior of gas flow rate and distributions of gas saturation, pressure, and permeability subject to different conditions and the stated mechanisms is explored. Water blockage reduces gas relative permeability over a large zone and reduces the gas flow rate. Despite gas flowing, strong capillary forces sustain mobile water over the entire system. Reducing drawdown gave lower driving force and higher resistance (by water blockage) for gas flow. The results show that 75% reduction of drawdown made the gas flow rate a couple orders of magnitude lower compared to if there was no blockage. The impact was most severe in more water-wet systems. The blockage caused most of the pressure drop to occur near the outlet. High pressure in the rest of the system reduced effects from gas decompression, matrix compression, and slip-enhanced permeability, whereas rapid gradients in all these effects occurred near the outlet. Gas decompression resulted in an approximately 10 times higher Darcy velocity and pressure gradient near the outlet compared to inlet, which contributed to removing blockage, but the added resistance reduced the gas production rate. Similarly, higher gas Corey exponent associated gas flow with higher pressure drop. The result was less blockage but lower gas production. Slip increased permeability, especially toward the outlet, and contributed to increase in gas production by 16%. Significant matrix compression was associated with permeability reduction and increased Corey exponent in some examples. These effects reduced production and shifted more of the pressure drop toward the outlet. Upstream pressure was more uniform, and less compression and permeability reduction were seen overall compared to a system without water blockage.acceptedVersio

Comparison of intercept methods for correction of steady state relative permeability experiments for capillary end effects

Steady-state relative permeability experiments are performed by coinjection of two fluids through core plug samples. The relative permeabilities can be calculated using Darcy’s law from the stabilized pressure drop and saturation of the core if capillary end effects and transient effects are negligible. In most cases, such conditions are difficult to obtain. Recent works have presented ways to extrapolate steady-state pressure drop and average saturation measurements affected by capillary end effects collected at different rates to obtain correct relative permeabilities at correct saturations. Both the considered methods are based on linear extrapolations to determine intercepts. Gupta and Maloney (2016) derived their method intuitively and validated it with numerical and experimental data. Andersen (2021a) derived a method from fundamental assumptions and presented an intercept method in a different form where the saturation and relative permeabilities are found directly and uniquely from straightline intercepts. All system parameters, including saturation functions and injection conditions, appear in the model. In this work, the two methods are compared. It is proven theoretically that Gupta and Maloney’s method is correct in that it produces the correct saturation and pressure drops corrected for capillary end effects. Especially, a constant pressure drop was assumed and here proved to exist, as a result of capillary end effects in addition to the Darcy law pressure drop with no end effects. Their method assumes a well-defined end effect region with length xCEE, but this length can be defined almost arbitrarily. This choice has little impact on average saturation and pressure drop, however. They also assumed that for a defined end effect region, the average saturation was constant and equal to the slope in their saturation plot. It is shown that if the region is defined, the average saturation is indeed constant, but not given by the slope. The correct slope is predicted by the Andersen model. We also comment on theoretical misinterpretations of the Gupta and Maloney method. A few works have correctly calculated that the pressure drop over the end effect region is independent of rate, but not accounted for that its length is rate dependent. We show that the combined pressure drop is equal to a constant plus the Darcy pressure drop over the full core. Examples are presented to illustrate the model behaviors. Literature datasets are investigated showing that (a) apparently rate-dependent CO2-brine relative permeability endpoints can be explained by capillary end effects and (b) the intercept methods can be applied to correct shale relative permeabilities.acceptedVersio

Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Time Using Physics-Informed Neural Networks

The application of physics-informed neural networks (PINNs) is investigated for the first time in solving the one-dimensional countercurrent spontaneous imbibition (COUCSI) problem at both early and late time (i.e., before and after the imbibition front meets the no-flow boundary). We introduce the utilization of Change-of-Variables as a technique for improving the performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables. The first describes saturation as a function of normalized position X and time T; the second as a function of X and Y = T0.5; and the third as a sole function of Z = X/T0.5 (valid only at early time). The PINN model was generated using a feed-forward neural network and trained based on minimizing a weighted loss function, including the physics-informed loss term and terms corresponding to the initial and boundary conditions. All three formulations could closely approximate the correct solutions, with water saturation mean absolute errors around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at early time. The Z formulation perfectly captured the self-similarity of the system at early time. This was less captured by XT and XY formulations. The total variation of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. Redefining the problem based on the physics-inspired variables reduced the non-linearity of the problem and allowed higher solution accuracies, a higher degree of loss-landscape convexity, a lower number of required collocation points, smaller network sizes, and more computationally efficient solutions.publishedVersio

Simulation study of wettability alteration enhanced oil recovery during co-current spontaneous imbibition

Naturally fractured reservoirs are highly dependent on capillary forces to recover hydrocarbons during water injection. Water can spontaneously imbibe and expel oil if positive capillary forces exist; purely counter-current if all sides of the matrix blocks are exposed to water; and predominantly co-current with some counter-current production if the blocks are exposed to water and oil simultaneously. The latter is referred to as a co-current spontaneous imbibition (SI) setup. Wettability alteration (WA) has been identified as a key mechanism to improve oil recovery from naturally fractured reservoirs, however almost all experimental and modeling studies on WA during SI have focused on counter-current SI. Our review indicates limited systematic experimental work on co-current SI using nonzero initial saturation, mixed wettability or WA processes and This modeling study will investigate enhanced oil recovery by WA during co-current SI where a brine with a general WA component imbibes and causes the system to become more water-wet. We model a 1D oil-saturated core exposed to water at one end (inlet) and oil at the other end (outlet), thus facilitating co-current SI. The core is initially preferentially (not strongly) oil-wet with low SI potential. The component is both transported by the imbibing brine and diffuses towards the imbibition front. Adsorption of the component is assumed to improve the water-wetness of the porous medium and hence the SI potential. The model is parameterized using consistent capillary pressure and relative permeabilities from previous history matching of brine-dependent porous disc experiments. The behavior of co-current SI at mixed-wet state is examined and compared to that of literature strongly water-wet behavior. Both secondary and tertiary enhanced recovery by SI with WA component is then considered in the simulations. Important parameters such as mobility ratio (as via oil viscosity), capillary back pressure, WA component concentration, adsorption and time of WA component exposure are investigated. Under mixed-wet conditions, favorable and unfavorable mobility ratios do not limit oil production as can be the case for strongly wetted media at unfavorable mobility ratio. This is due to oil preserving mobility at all obtained saturations. A third or more of the total production was counter-current, which is high compared to strongly wetted media. It was shown that half the oil could be produced counter-currently as an upper limit. High oil mobility is preserved in the twophase region near the inlet and was found to ensure a high minimum fraction of counter-current production. Twice as much of the incremental oil from WA was produced counter-currently as co-currently, explained by increased oil relative permeability in the WA affected inlet region. Sensitivity analysis revealed that an opposite shift would reduce the incremental counter-current production despite raised local capillary forces. Capillary back pressure resists oil production at the inlet without limiting water from imbibing. As a result, capillary back pressure had significant impact on co-current SI simulations with fixed and changing wettability. The trends discovered in this study, both for mixed-wet and wettability alternating systems, are hoped to inspire future experimental measurements.acceptedVersio

Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)

In recent years, the gap between Deep Learning (DL) methods and analytical or numerical approaches in scientific computing is tried to be filled by the evolution of Physics-Informed Neural Networks (PINNs). However, still, there are many complications in the training of PINNs and optimal interleaving of physical models. Here, we introduced the concept of Physical Activation Functions (PAFs). This concept offers that instead of using general activation functions (AFs) such as ReLU, tanh, and sigmoid for all the neurons, one can use generic AFs that their mathematical expression is inherited from the physical laws of the investigating phenomena. The formula of PAFs may be inspired by the terms in the analytical solution of the problem. We showed that the PAFs can be inspired by any mathematical formula related to the investigating phenomena such as the initial or boundary conditions of the PDE system. We validated the advantages of PAFs for several PDEs including the harmonic oscillations, Burgers, Advection-Convection equation, and the heterogeneous diffusion equations. The main advantage of PAFs was in the more efficient constraining and interleaving of PINNs with the investigating physical phenomena and their underlying mathematical models. This added constraint significantly improved the predictions of PINNs for the testing data that was out-of-training distribution. Furthermore, the application of PAFs reduced the size of the PINNs up to 75% in different cases. Also, the value of loss terms was reduced by 1 to 2 orders of magnitude in some cases which is noteworthy for upgrading the training of the PINNs. The iterations required for finding the optimum values were also significantly reduced. It is concluded that using the PAFs helps in generating PINNs with less complexity and much more validity for longer ranges of prediction.Comment: 26 pages, 9 figure

Theoretical Comparison of Two Setups for Capillary Pressure Measurement by Centrifuge

There are several approaches for the calculation of capillary pressure curves in porous media including the centrifuge method. In this work, a new installation of centrifuge test is introduced and compared with the traditional setup. In the first setup, which is a standard approach in labs, the core face closest to the rotational axis is open to the non-wetting phase, while the farthest face is open to the wetting phase where strictly co-current flow is generated in rotations; labeled Two-Ends-Open (TEO). In the second setup, which is proposed as a new approach, only the outer radius surface is open and is exposed to the light non-wetting phase; labeled One-End-Open (OEO). This setup strictly induces counter-current flow. The two systems and their corresponding boundary conditions are formulated mathematically and solved by a fully implicit numerical solver. The TEO setup is validated by comparison with commercial software. Experimental data from the literature are used to parameterize the models. It is mathematically, and with examples, demonstrated that the same equilibrium is obtained in both systems with the same rotational speed, and changing the installation does not influence the measured capillary pressure. This equilibrium state is only dependent on the rotational speed, rock capillary pressure properties, and fluid densities, not the installation geometry, relative permeabilities, or fluid viscosities. However, the dynamic transition trend and saturation profiles were found to be dependent on the applied installation. It was observed that the OEO setup takes almost identical equilibration time as the TEO setup for mixed-wet states, although it needed much longer time in water-wet states. The presence of threshold capillary pressure significantly increased the time scale of the OEO setup. Also, it was found that in contradiction to the TEO setup, the dynamic saturation profile in OEO was rarely influenced by viscosity ratio. To conclude, the performed history matching analysis demonstrated that the OEO setup can be applied for the calculation of counter-current relative permeability from the production data with reasonable accuracy.publishedVersio