Prediction of relative and absolute permeabilities for gas and water from soil water retention curves using a pore‐scale network model

Relative permeability of the nonwetting phase in a multiphase flow in porous media is a function of phase saturation. Specific expressions of this function are commonly determined by combining soil water retention curves with relative nonwetting phase permeability models. Experimental evidence suggests that the relative permeability of the nonwetting phase can be significantly overestimated by the existing relative permeability models. A new model for the prediction of relative nonwetting phase permeability from soil water retention curves is proposed in this paper. A closed form expression can be obtained in combination with soil water retention curves. The model is mathematically simple and can easily and efficiently be implemented in numerical models of multiphase flow processes in porous media. The predicting capability of the proposed model is contrasted with well‐supported models by comparing the measured and predicted relative air permeability data for 11 soils, representing a wide range of soil textures, from sand to silty clay loam. In most of the cases the proposed model improves the agreement between the predicted relative air permeability and the measured data.

The objective of this study is to define the physical basis approach to calculate relative permeability of reservoir rocks from image analysis of thin section. Relative permeability is one of the most important parameter and widely used for reservoir characterization. Relative permeability is only measured in the laboratory. To measure relative permeability in the laboratory are expensive and time consuming. Alternatively relative permeability may be estimated from numerical methods such as Lattice-Boltzmann and network modeling. To applying the all method need lot of computational effort and the physical justification of relative permeability estimation remains hidden. A simple and powerful equation to estimate the absolute permeability from porosity and specific surface area is the Kozeny’s equation. Darcy’s assumption was applied to Kozeny’s equation to calculate relative permeability of two phase-fluid flow without introducing any empirical factor. Numerical simulation and image analysis processing technique were applied to check the consistency of Kozeny’s approximated relative permeability model. Relative permeability predicted from Kozeny’s approximation was compared with permeability predicted from Lattice-Boltzmann flow simulation as well as with laboratory measured data. Relative permeability predicted from numerical simulation and image analysis techniques are well agreed with permeability predicted from Kozeny’s approximation. Relative permeability predicted from Kozeny’s approximation are also comparable with permeability predicted from Lattice-Boltzmann flow simulation as well as with laboratory measured data. Given permeability model in this study is the physical basis approach and may used to calculated relative permeability from image analysis of thin section.

This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation, and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The three-phase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coefficients, absolute permeability, and relative permeability fields are estimated by automatic history matching of production data. Introduction The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problemunder three-phase flow conditions. It appears that Archer and Wong(1) were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching. Sigmund and McCaffery(2) were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson(3) considered a similar problem. They calculated predicted data from the Buckley- Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to those obtained using a power law functional form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known. Lee and Seinfeld(4) considered the simultaneous estimation of the absolute permeability field and relative permeabilities for a two-dimensional, two-phase flow oil-water system. They assumed power law relative permeability curves and assumed that the end point values of relative permeabilities were known. Thus, only the two exponents in the power law relative permeability functions were estimated. They modelled the two-dimensional isotropic heterogeneousermeability field using bi-cubic B-splines. In the specific xamples considered, they matched pressure and water cut data at wells producing from an oil reservoir under waterflood. Tikhonov(5) regularization was used to stabilize the nonlinear least squares problem.

Investigating NMR-based absolute and relative permeability models of sandstone using digital rock techniques

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower in the presence of gas during waterflooding due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase. We show how to use network modeling to develop a physically-based empirical model for three-phase relative permeability. We demonstrate that the relative permeabilities are approximately independent of saturation path when plotted as a function of flowing saturation. The flowing saturation is the saturation minus the amount that is trapped. The amount of oil and gas that is trapped shows a surprising trend with wettability - weakly water-wet media show more trapping of oil and gas than a water-wet system due to the complex competition of different three-phase displacement processes. Further work is needed to explore the full range of behavior as a function of wettability and displacement path.

Permeability anisotropy of methane hydrate-bearing sands: Insights from CT scanning and pore network modelling

Positive velocity dependency of relative permeability of gas–condensate systems, which has been observed in many different core experiments, is now well acknowledged. The above behaviour, which is due to two-phase flow coupling in condensing systems at low interfacial tension (IFT) conditions, was simulated using a 3D pore network model. The steady-dynamic bond network model developed for this purpose was also equipped with a novel anchoring technique, which was based on the equivalent hydraulic length concept adopted from fluid flow through pipes. The available rock data on the co-ordination number, capillary pressure, absolute permeability, porosity and one set of measured relative permeability curves were utilised to anchor the capillary, volumetric and flow characteristics of the constructed network model to those properties of the real core sample. Then the model was used to predict the effective permeability values at other IFT and velocity levels. There is a reasonable quantitative agreement between the predicted and measured relative permeability values affected by the coupling rate effect.

This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The threephase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-dimensional, three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coeffcients, absolute permeability and relative permeability fields are estimated by automatic history matching of production data. To the best of our knowledge, this is the first work which considers the simultaneous estimation of heterogeneous permeability fields and three-phase relative permeability curves by the automatic history matching of three-phase ow production data. Introduction The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problem under three-phase flow conditions. It appears that Archer and Wong1 were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching. Sigmund and McCaffery2 were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson3 considered a similar problem. They calculated predicted data from the Buckley-Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to these obtained using a power law function form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known. In a later paper, the same authors 4 showed how to impose constraints to ensure that the estimated relative permeabilities are concave up (convex downward), nonnegative and monotonic. The Levenberg-Marquardt modification of the Gauss-Newton method was applied for optimization. With cubic spline representations, not all coefficients are independent. Kerig and Watson3 presented a procedure to determine the parameters that should be adjusted when matching core flood data.

Summary A first attempt has been made to predict three-phase relative permeability experimental data of a water-wet Berea sandstone obtained by Oak (1990) using the three-phase flow network model for arbitrary wettability developed by van Dijke and Sorbie (2002a). First, the network model is anchored to the corresponding two-phase relative permeability and capillary pressure data using an idealized representation of the pore geometry and a simple parameter-fitting procedure. Then, predictions of three-phase properties are made, which are compared with experimental data as well as previous predictions from different network models. The present study has confirmed that the relatively simple network model, anchored to experimental data, is able to predict three-phase relative permeabilities with reasonable accuracy, comparable to the accuracy of more-complex models. On the basis of these preliminary results, a limited sensitivity study is carried out with respect to different wettability states and two combinations of interfacial tensions (IFTs). This study reveals some new results with respect to the invariance of relative permeability to interfacial-tension combinations and the trend of water relative permeability as a function of the fraction of oil-wet pores in systems of nonuniform wettability.

A first attempt has been made to predict three-phase relative permeability experimental data of a water-wet Berea sandstone obtained by Oak1 using the three-phase flow network model for arbitrary wettability developed by van Dijke and Sorbie2. First, the network model is anchored to the corresponding two-phase relative permeability and capillary pressure data using an idealised representation of the pore geometry and a simple parameter fitting procedure. Then, predictions of three-phase properties are made, which are compared with experimental data as well as previous predictions from a different network models. The present study has confirmed that the relatively simple network model, anchored to experimental data, is able to predict three-phase relative permeabilities with reasonable accuracy, comparable to the accuracy using more complex models. Based on these preliminary results a limited sensitivity study is carried out with respect to different wettability states and two combinations of interfacial tensions. This study reveals some new results with respect to the invariance of relative permeability to interfacial tension combinations and the trend of water relative permeability as a function of the fraction of oil-wet pores in systems of non-uniform wettability.

Secondary oil recovery by waterflooding is usually achieved in neutrally and water wet reservoirs. Special core analysis (SCAL) data are difficult to obtain and expensive, we therefore generate experimentally-derived petrophysical correlations based on SCAL experiments conducted on neutrally wet sandstone in order to further understand maximizing oil recovery. The key objectives of the present study are to obtain and analyze the possible relationships between the reservoir petrophysical properties that are essential for reservoir simulations. The experimentally obtained petrophysical parameters and generated relationships are compared versus the prevailing initial water saturation, absolute permeability and wettability. Experimental results are then compared with network modeling results from the literature. Network modeling is a promising tool that has many usages including the convenient estimation of capillary pressure, relative permeability and residual oil saturation that would otherwise be obtained through lengthy and expensive SCAL experiments. However, network modeling predictively has been questioned in terms of making a full priori prediction of multiphase flow properties in mixed and neutrally wet systems as true representation of pore geometry and wettability are most challenging. Scaled, two-phase oil-water system, endpoint relative permeability was found to be linearly and strongly correlated to wettability. Residual oil saturation, however, was found to be curvilinear upward correlated to Amott-Harvey wettability. Comparing the experimental results of the present study with previous network model results, limited agreement was observed. Disagreement was mainly due to pore space wettability misrepresentation in network models. The representation of neutral wettability is the path towards more realistic physical description of pore-scale multiphase flow since most reservoirs tend to show some degree of neutral wettability.

A pore-scale model consisting of a network of pore bodies inter-connected by pore throats is used to calculate drainage relative permeabilities and capillary pressure for a strongly water wet Berea sandstone core. The architecture and geometry of the pore network which is used in the model is constructed from thin section analysis and numerical modelling of the main sandstone-forming geological processes, i.e., grain sedimentation, compaction, and diagenesis. The effect of different pore network descriptors on relative permeability at low capillary numbers has been simulated. The results show that pore shapes strongly influence wetting phase relative permeability, particularly at low saturations where film flow is important. Simulated relative permeabilities are found to be in good agreement with those predicted from an empirically derived correlation. Introduction Macroscopic multiphase flow in porous media is usually described in terms of Darcy's law and measured or empirically derived saturation dependent relationships for phase relative permeabilities and capillary pressure. Accurate and consistent acquisition and interpretation of such data are essential for almost all reservoir engineering calculations and determine to a large extent how reservoir management can optimise oil production and recovery. Relative permeability measurements, either by steady state or unsteady state methods, are time consuming, expensive, and often difficult to interpret. As a result, too few measurements are usually performed and numerous uncertainties may be associated with the measurements. This prohibits assigning unique relative permeability functions to different architectural units in the reservoir (i.e., channels, crevasse splays, wash-over fans, etc.) and limits the ability of reservoir simulators to accurately predict oil recovery. Relative permeabilities and capillary pressure are averaged transport properties which represent the physical processes occurring on the pore-scale. On the pore-scale, the displacement of one fluid by another is controlled by interfacial tension, viscous forces, rock-fluid interactions, and the geometry of the pore space. In principle, it should therefore be possible to determine relative permeabilities and capillary pressure by appropriately averaging the equations describing the physical processes occurring on the microscopic or pore-scale. This approach requires a detailed understanding of the displacement mechanisms on the pore-scale and a complete description of the morphology of the pore space. The procedure has successfully been applied to two- and three-phase flow in simple or idealised porous media using pore-scale physics identified in micromodel experiments with the morphology of the pore space represented by a topologically equivalent numerical network. The difficulty in constructing a realistic three-dimensional (3-D) representation of the complex pore structure of real porous rocks has limited the above approach to idealised porous media. Although advanced techniques such as serial-sectioning and micro-CT are available, information about the pore structure of porous rocks is usually obtained from image analysis of 2-D thin section images of rocks and from mercury injection capillary pressure curves. Thin sections provide aerial information which is relevant to porosity measurements whilst mercury injection data provide information about the volume of pores which may be invaded through pore throats within specified size ranges. These data are insufficient to provide a complete 3-D description of the architecture and geometry of the pore space and do not allow construction of 3-D pore networks which accurately represent the complex pore space of a given porous rock. In the present work, stochastic modelling of the main sandstone-forming geological processes (i.e., sedimentation, compaction, and diagenesis) are combined with 3-D image analysis techniques to generate a realistic and fully characterised 3-D representation of the pore network for a Berea sandstone. P. 345

Relative permeabilities are complex and important rockfluid properties of reservoirs in which multi-phase flow conditions prevail. Measuring relative permeabilities in the laboratory using cores obtained from a reservoir is a complicated, time demanding and labor-intensive task. There has been limited success in mathematical modeling of relative permeabilities based on rocks and fluid properties due to our inability to simulate the non-linear controlling mechanisms in place. Artificial neural networks (ANNs) promise a potential avenue for implicitly incorporating the controlling mechanisms and parameters into a model that can be utilized as an effective tool for relative permeability predictions. The methodology described in this paper exploits the unique topology of ANNs for determining the two-phase (oilwater) relative permeabilities. The ANN is a universal approximator that performs non-linear, multi-dimensional interpolations. in the development stage of the ANN model, a large number of oil-water relative permeability data sets were collected from the literature. These data sets were used to train the model. In composing the architecture of the ANN, only the readily available rock and fluid properties (endpoint saturations, porosity, permeability, viscosity, and interfacial tension) have been explicitly incorporated. The predictive ability of the model was tested using experimental data sets that were not used during the training stage. The results are in good agreement with the experimentally reported data. The proposed model exhibits sensitivity to several reservoir properties. The proposed ANN model has a dynamic training base that can be expanded as new data become available. Introduction Relative permeabilities quantify multi-phase flow through porous media. Generating an accurate relative permeability versus saturation relationship for each phase is essential for evaluating the performance of a reservoir during primary, secondary, and tertiary production periods. Relative permeability-saturation relationships vary between reservoirs and within a given reservoir. They are non-linear functions of reservoir rock and fluid properties such as phase saturations, formation types, depositional environment, shale content, heterogeneity, porosity, permeability, interconnectivity of pores, pore geometry, interfacial tensions between flowing phases, phase viscosities, phase densities, and rock wettability. Experimental and modeling methods are used for assigning relative permeabilities to a reservoir. Although laboratory measurements of relative permeabilities are difficult, they are still the preferred method. Laboratory measurements are technically difficult and require skillful personnel, expensive equipment, and are lengthy to perform. Therefore, estimation of relative permeabilities with mathematical models has always been an attractive goal. The accuracy of relative permeability values must be preserved. Relative permeability models are commonly used only as estimation tools. Mathematical models for relative permeability can be classified under four main categories: capillary, statistical, empirical, and network models. These models require some rock and fluid properties such as endpoint saturation values, porosity, absolute permeability, interfacial tensions, and viscosity of phases, and incorporate important assumptions. The models are restricted by their assumptions, are not universally applicable, and may be difficult to update for different systems. Empirical models and pore-network models are frequently used and are the most successful in estimating relative permeabilities. Predicting the relative permeability values using mathematical models is

MEMBERS SPE-AIME Abstract Equipment was constructed to perform dynamic displacement experiments on small core samples under conditions of elevated temperature. Oil-water flowing fraction and pressure drop were recorded continuously for calculation of both the relative permeability ratio and the individual relative permeability ratio and the individual relative permeabilities. Imbibition relative permeabilities permeabilities. Imbibition relative permeabilities were measured for five samples of Boise sandstone at room temperature and at 175deg.F. The fluids used were distilled water and a white mineral oil. The effect of temperature on absolute permeability was investigated for six Boise sandstone samples and two Berea sandstone samples. Results for all samples were similar. The irreducible water saturation increased significantly, while the residual oil saturation decreased significantly with temperature increase. The individual relative permeability to oil increased for all water saturations below the room-temperature residual oil saturation, but the relative permeability to water at flood-out increased with permeability to water at flood-out increased with temperature increase. Absolute permeability decreased with temperature increase. Introduction Test environment is generally acknowledged to have a significant effect on measurement of relative permeability. The environment consists not only permeability. The environment consists not only of the temperature and pressure, but also of the fluids used and the core condition. Several workers have used the approach of completely simulating the reservoir conditions in the laboratory experiment. Such methods are termed "restored state." Restored state data are generally different from "room condition" data; since several variables are involved, it is difficult to determine the importance of each variable. Another approach used attributes the changes in relative permeability to changes in the rock-fluid interaction or wettability. Wettability, however, depends on many variables. Specifically, wettability depends on the composition of the rock surface, the composition of the fluids, the saturation history of the rock surface, and the temperature and pressure of the system. The purpose of this study is to isolate temperature as a variable in the relative permeability of a given rock-fluid system. Work on isolation of temperature as a variable in relative permeability has been conducted since the early 1960s. Edmondsons established results in 1965 for a Berea sandstone core using both water/refined oil and water/crude oil as fluid pairs. He showed a change in the relative permeability ratio accompanied by a decrease in the residual oil saturation with temperature increase. Edmondson showed no data for water saturations below 40 percent, and his curves show considerable scatter in the middle saturation ranges. Edmondson's work was the only study to use consolidated cores to investigate the effect of temperature on relative permeability measurements. Poston et al. presented waterflood data for sand packs containing 80-, 99-, a nd 600-cp oil, and packs containing 80-, 99-, a nd 600-cp oil, and observed an increase in the individual relative permeabilities with temperature increase. The permeabilities with temperature increase. The increase in the oil and the water permeability was accompanied by an increase in irreducible water saturation and a decrease in the residual oil saturation with temperature increase. Poston et al. was the only work to present individual oil and water permeability. Davidsons presented results for displacement of No. 15 white oil from a sand pack by distilled water, steam, or nitrogen. However, he found little permeability-ratio dependence in the middle permeability-ratio dependence in the middle saturation ranges. Davidson, too, found a decrease in the residual oil saturation with temperature increase, but he did not include data on irreducible water saturation. SPEJ P. 376

This paper describes a new method to estimate two- and three-phase relative permeabilities in-situ, using pressure transient analysis. The technique requires a short drawdown test, consisting of a number of steps of increasing flow rate. The resulting relative permeabilities reflect the properties of the whole drainage area, rather than those of a small laboratory core. The proposed technique is a major improvement over current historical performance methods. These existing methods need data over long periods, yet only cover a range of saturation up to present conditions – all future projections require extrapolation. By contrast, the new method estimates relative permeabilities at sandface saturations, which cover a range of future reservoir conditions. The well test can be repeated at a later stage of depletion to forecast still further into the future. The proposed technique applies the solutions of multiphase diffusivity equation in terms of the pseudopressure function, m(p). These solutions have already been reported for constant rate tests in solution gas-drive reservoirs. This paper extends the pseudopressure solutions to three-phase systems. Two- and three-phase solutions are then superposed to obtain multiple-rate solutions, the basis for two- and three-phase relative permeability equations. A saturation equation developed originally by Bϕe et al. for solution gas-drive reservoirs, is also extended here to three-phase reservoirs. These pressure-saturation equations can be used to estimate sandface saturations during the test. Using a commercial black-oil simulator, example well tests were simulated over a 40% range in gas saturation. Both two- and three-phase results show close agreement with input relative permeability curves. In cases where relative permeability is not homogeneous within the drainage area, the resulting estimates of relative permeability curves were very representative of in-situ heterogeneities. The proposed technique offers a means to estimate two- and three-phase relative permeabilities at in-situ reservoir conditions, accounting for heterogeneities, wettability and fluid composition. It also seems to be the most appropriate way to obtain estimates for subsequent reservoir engineering analysis.