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In the process of coalbed methane (CBM) exploitation, permeability is a key controlling parameter for gas transport in the CBM reservoirs. The CBM reservoir contains a large number of micropores and microfractures with complex structures. In order to accurately predict the gas permeability of the micropores and microfractures in CBM reservoirs, a fractal permeability model was developed in this work. This model considers the comprehensive effects of real gas, stress dependence, multiple gas flow mechanisms (e.g., slip flow, Knudsen diffusion and surface diffusion) and fractal characteristics (e.g., pore size distribution and flow path tortuosity) of micropores and microfractures on the gas permeability. Then, this fractal permeability model was verified by the reliable experimental data and other theoretical models. Finally, the sensitivity analysis is conducted to identify key factors to the permeability of CBM reservoir. The results showed that the fractal characteristics of the micropores and microfractures have significant effects on the permeability of CBM reservoir. Higher fractal dimension of micropores diameter and microfractures aperture represents the larger number of micropores and microfractures, resulting in a higher permeability. Higher tortuosity fractal dimension of micropores and microfractures means higher gas flow resistance, leading to the lower permeability. The multiple gas transport mechanisms coexist in micropores and microfractures of CBM reservoir. The permeability of slip flow and Knudsen diffusion both increases with the decrease in pore pressure. Surface diffusion is an important gas transport mechanisms in micropores, but it can be ignored in the microfractures. Knudsen diffusion plays a more obvious role in the lower pore pressure, which controls the gas transport in microfractures. And microfractures are beneficial to improve gas transport capacity of coalbed methane reservoir.

In this work, the tortuosity and the permeability of a porous medium whose geometry is idealized according to two Euclidean spatial dimensions is studied. Unit porous media solid matrices are taken as iterations toward the 2D Sierpinski carpet fractal geometry. Investigated porous media unit arrangements are either fully periodic (FP), or that of a periodic channel (PC), in which unit porous media geometries are infinitely stacked on both orthogonal directions or only in the flow direction inside a walled channel, respectively. A third case called adjusted fully periodic was also proposed as a counterpoint to the formation of the single preferential flow path of the FP case. The flow regime of interest is the Stokes (or creeping) one, and the numerical approach is done using the lattice Boltzmann method on a porous medium unit domain with a preset pressure drop. Moreover, the scale analysis technique is applied to obtain theoretical correlations for the permeability as a function of porous medium properties for the three cases. Good agreement is found between the correlations obtained herein with results available in the literature. A finding is that whenever the channel walls are removed, the PC case correlation recovers the one for FP. Finally, the results suggest a limit value for the iteration of the carpet beyond which the tortuosity and the permeability of the porous channel and the infinitely periodic (unwalled) porous media are equal.

International audience; Electrokinetic and electroosmotic couplings can play important roles in water and ions transport in charged porous media. Electroosmosis is the phenomena explaining the water movement in a porous medium subjected to an electrical field. In this work, a new model is obtained through a new up-scaling procedure, considering the porous medium as a bundle of tortuous capillaries of fractal nature. From the model, the expressions for the electroosmosis pressure coefficient, the relative electroosmosis pressure coefficient, the maximum back pressure, the maximum flow rate, the flow rate-applied back pressure relation and the product of the permeability and formation factor of porous media are also obtained. The sensitivity of the relative electroosmosis pressure coefficient is then analyzed and explained. The model predictions are then successfully compared with published datasets. Additionally, we deduce an expression for the relative streaming potential coefficient and then compare it with a previously published model and experimental data from a dolomite rock sample. We find a good agreement between those models and experimental data, opening up new perspectives to model electroosmotic phenomena in porous media saturated with various fluids.

Permeability of fractured media plays an important role in many hydrogeologic applications. In this study, we propose an approach to quantify the equivalent permeability of fractal fractured media along with the fracture length direction, and both the laminar flow and nonlinear flow are considered according to the fracture features and critical Reynolds number. The analytical expression for equivalent permeability of fractured media is derived, which depends on fractal fracture properties and the applied conditions. The new approach is applicable for any Newtonian fluid. The effects of fractal dimension for aperture and length, maximum fracture aperture, ratio of minimum to maximum aperture, critical Reynolds number and pressure head loss across the fractures are explored and discussed. The fractal dimension for fracture aperture is positively correlated with the porosity of porous media, and a clear power-law correlation between the equivalent permeability and porosity can be observed in the results. The maximum aperture and the ratio of minimum to maximum aperture control the size of fracture apertures, and the increase in both parameters leads to high permeability. The larger fractal dimension for length causes more tortuous fractures, which reduce the equivalent permeability. A higher critical Reynolds number leads to a larger equivalent permeability. A larger pressure head loss results in a smaller equivalent permeability.

Oil migration processes are crucially important for determining oil reserve levels and distributions in low-permeability reservoirs. Previous researches have proved that the macroscopic parameters of oil migration processes are controlled by microscopic pore properties in low-permeability media. To analyze the relationship between pore-scale parameters and the oil migration process in low-permeability formations, a mathematical model combining the capillary bundle model and fractal theory is developed in this work. The accuracy of the proposed model is validated via comparisons with three well-designed oil charging experiments using natural core samples. Based on the validated model, the influence of four factors (pore fractal dimension, tortuosity fractal dimension, the wettability of the rock surface, and the formation water viscosity) on oil migration processes in low-permeability media is analyzed. Oil saturation and effective permeability are used as the output parameters for reflecting changes in the oil migration process. The calculation results indicate that oil saturation decreases as pore fractal dimension, tortuosity fractal dimension, the hydrophilicity of the rock surface, and brine viscosity increase. On the other hand, effective permeability decreases as tortuosity fractal dimension and formation water viscosity increase, but increases with pore fractal dimension. Moreover, the wettability dependence of effective permeability is relatively weak.

In order to study gas transport properties of fractured shale gas reservoirs for the accurate estimation of shale gas production, a new multiscale fractal transport model with an effective porosity model was proposed based on the fractal theory and the multilayer fractal Frenkel–Halsey–Hill (FHH) adsorption. In shale matrix, both fractal microstructures of pores (such as pore size distribution, flow path tortuosity, and pore surface roughness) and multiscale flow mechanisms (including slip flow and Knudsen diffusion) were coupled. In fracture network, fractal fracture length distribution, stress compaction, and gas pressure were introduced to formulate a new fracture permeability model. These permeability and effective porosity models were then incorporated into the governing equations of gas flow and the deformation equation of reservoirs to form a numerical model. This numerical model was solved within COMSOL Multiphysics for shale gas recovery. Both transport models in shalematrix and fracture network were validated by experimental data or compared with other models. Finally, sensitivity analysiswas conducted to identify key parameters to gas recovery enhancement. Itwas found that themultilayer gas adsorption and fractal microstructures have great impacts on gas production in shale reservoirs. The cumulative gas production can be increased by 26% after 8000 days when themultilayer adsorbed gas is considered. Larger surface fractal dimension and larger tortuosity fractal dimension represent more roughness pore surface, higher flow resistance, and lower cumulative gas production. Bigger pore diameter fractal dimension means more pores, higher permeability, and higher cumulative gas production. Our model with fractal FHH adsorption was in better agreements with field data from Marcellus and Barnett shale reservoirs than other models.

Permeability is one of the key parameters for quantitatively evaluating groundwater resources and accurately predicting the rates of water inflows into coal mines. This paper presents an efficient method to estimate the macroscopic permeability by using the scanning electron microscopy (SEM) images. A correlation between the microscopic features of sandstone porosity and the macropermeability is approached by an image identification technique. Firstly, the gray images were transformed into the binary images by using the histogram of the entropy method. Then, the Green and Euler distance methods were applied to calculate the length and area of the pores, and the fractal parameters were estimated according to the slit island method. Based on the theory of microscopic seepage flow, the seepage coefficient and permeability were calculated by fractal parameters. Typical water-bearing sandstone samples in the Kailuan coal field area in North China were selected to demonstrate the methodology. SEM microscopic images of nine groups of sandstone samples collected at different depths (from the outcrop to the deep mines) were analyzed. Based on the theoretical model of micropore structures and the fractal theory, the permeabilities were estimated. The results provide insights for understanding the hydraulic properties of the sandstone.

To evaluate pore structures of the Bakken Shale, which is one of the most important factors that affect petrophysical properties, high-pressure mercury intrusion was employed in this study. Pore structures such as pore-throat size, pore-throat ratio, and fractal attributes are investigated in this major shale play. Pore-throat size from 3.6 to 200 um is widely distributed in these shale samples. Accordingly, pore-throat size distributions demonstrate the multimodal behavior within the samples. The whole pore-throat network can be divided into four clusters: one set of large pores, two transitional/intermediate pore groups, and one set of smaller pores. The fractal analysis revealed that fractal dimensions decrease as the pore-throat size decreases. The multifractal analysis demonstrated that as the maturity of the shale samples increases, pore-throat size distributions would become more uniform and pore structures tend to become more homogeneous. The results are compared to our previous results obtained from nitrogen gas adsorption for further verifications of fractal behavior. Finally, although fractal analysis of mercury intrusion and nitrogen gas adsorption were comparable, the results of multifractal analysis from these two methods were not identical.

Coal seam water infusion is a universal comprehensive mine disaster prevention method practiced worldwide. The result of water infusion is determined by the structure and seepage characteristics of the coal sample around the borehole. In this paper, the structure and seepage characteristics of the coal sample under various stress and pore water pressures are measured via nuclear magnetic resonance (NMR) technology; the influence rule of confining pressure and pore water pressure during the test on the coal sample structure and seepage characteristics is analyzed. Based on fractal geometrical theory, the fractal model of permeability is created. The results show that the coal sample with water infusion has an intertwined internal fracture-pore structure and that the pore radius distribution is diverse. Through theoretical analysis and discussion, we found that there are no large changes in the pore area fractal dimension and the measured pore volume fractal dimension, but the trends of changes in these two fractal dimensions are opposite, because the pore area fractal dimension characterizes the homogeneity of the pore area distribution on the surface of a coal sample, and the measured pore volume fractal dimension characterizes the cumulative volumetric changes in the pores inside a coal sample. The changes in these two fractal dimensions validate that the pore structures inside a coal sample have similar in fractal characteristics and demonstrate that the coal seam water infusion technique will not damage the skeleton of the coal sample. The variation rules of theoretical permeability from the fractal model and the value from the liquid measurement versus confining pressure and pore water pressure are consistent, an increase in the water pressure will result in an increase in the permeability, and an increase in the confining pressure will result in a decrease in the permeability. However, because the seepage channels with a large diameter in the tested coal sample were blocked, there is a relatively large difference between the two permeabilities. Therefore, increasing the connectivity between the seepage channels with a large diameter will improve the effects of water infusion, which is implemented to prevent disasters. Through NMR experiment and theoretical analysis, this study establishes a quantitative relationship between the pore structures inside the coal sample and its permeability during coal seam water infusion process, provides an advanced experimental approach and theoretical analysis method, which will be of great importance in the improvement in the water infusion process implemented in deep working coal seams to prevent disasters and in the determination of the range of application of this process and the evaluation metrics for this process.

Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions $$2.579 \le D_{\mathrm{m}} \le 2.893$$ characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension $$D_{\mathrm{s}}$$ and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing $$D_{\mathrm{s}}$$ from 1 to 1.1 regardless of $$D_{\mathrm{m}}$$ value, while k may slightly increase or decrease, depending on $$D_{\mathrm{m}}$$ , as $$D_{\mathrm{s}}$$ increases from 1.1 to 1.6.

Tight porous media are mainly composed of micro/nano-pores and throats, which leads to obvious microscale effect and nonlinear seepage characteristics. Based on the capillary bundle model and the fractal theory, a new nonlinear seepage equation was deduced, and a further fractal permeability model was obtained for oil transport in tight porous media by considering the effect of the boundary layer. The predictions of the model were then compared with experimental data to demonstrate that the model is valid. This model clarifies the oil transport mechanisms in tight porous media: the effective permeability is no longer a constant value and is governed by properties of tight porous media and oil. Furthermore, parameters influencing effective permeability were analyzed. The model can accurately present the seepage characteristics of the oil in tight porous media and provide a reliable basis for the development of unconventional reservoirs.

Tight and shale gas reservoirs are characterized by pores in the range of micro- and nanometer. The description of the gas flow in such tight porous media is complicated by the coexistence of various flow regimes. Generally, porous media is represented by a bundle of tortuous capillaries and contributions of various flow regimes are incorporated by numerical integration of the flow rate of individual capillary, which causes inconvenience for practical application. In this paper, approximate expressions for apparent gas permeability of tight porous media with a relative error less than 0.1 based on the lognormal pore size distribution and the fractal pore size distribution have been separately derived. With the proposed expressions, apparent gas permeability of tight porous media can be calculated without numerical integration. The proposed method has been validated by published data.

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

Among all properties of reservoir rocks, permeability is an important parameter which is typically measured from core samples in the laboratory. Due to limitations of core drilling all over a reservoir, simulation of rock porous media is demanded to explore more scenarios not seen in the available data. One of the most accurate methods is cross correlation based simulation (CCSIM) which recently has broadly applied in geoscience and porous media. The purpose of this study is producing realizations with the same permeability trend to a real sample. Berea sandstone sample is selected for this aim. Permeability results, extracted from smaller sub-samples of the original sample, showed that classic Kozeny–Carman permeability trend is not suitable for this sample. One reason can be due to lack of including geometrical and fractal properties of pore-space distribution in this equation. Thus, a general trend based on fractal dimensions of pore-space and tortuosity of the Berea sample is applied in this paper. Results show that direct 3D stochastic modeling of porous media preserves porous structure and fractal behavior of rock. On the other hand, using only 2D images for constructing the 3D pore structures does not reproduce the measured experimental permeability. For this aim, a hierarchical sampling is implemented in two and three steps using both 2D and 3D stochastic modeling. Results showed that two-step sampling is not suitable enough, while the utilized three-step sampling occurs to be show excellent performance by which different models of porous media with the same permeability trend as the Berea sandstone sample can be generated.

Optimising production from heterogeneous and anisotropic reservoirs challenges the modern hydrocarbon industry because such reservoirs exhibit extreme inter-well variability making them very hard to model. Reasonable reservoir models can be obtained using modern geostatistical techniques, but all of them rely on significant variability in the reservoir only occurring at a scale at or larger than the inter-well spacing. In this paper we take a different, generic approach. We have developed a method for constructing realistic synthetic heterogeneous and anisotropic reservoirs which can be made to represent the reservoir under test. The main physical properties of these synthetic reservoirs are distributed fractally. The models are fully controlled and reproducible and can be extended to model multiple facies reservoir types. This paper shows how the models can be constructed and how they have been tested. Reservoir simulation results of a number of generated 3-D heterogeneous and anisotropic models show that heterogeneity, in terms of only the geometric distribution of reservoir properties, has a little effect on oil production from high and moderate quality reservoirs. However, if the effect of heterogeneity on capillary pressure is taken into account, the effect becomes striking, where varying the heterogeneity of reservoirs properties can lead to a 70 % change in the predicted oil production rate and a significant early shift of water breakthrough time. Hence, it is the heterogeneity consequences that are really substantial if not taken into account. These are very significant uncertainties for a hydrocarbon company if the heterogeneity of their reservoir is not well defined.

Understanding multiphase flow in porous media is of tremendous importance for many industrial and environmental applications at various spatial and temporal scales. The present study consequently focuses on modeling multiphase flows by the volume-of-fluid method (sharp interface) in porous media with a simplified Darcy-scale approach and shows simulations of Saffman–Taylor fingering. The simplification of the Darcy-scale approach is performed by assuming sharp interfaces between pure phases. The volume-of-fluid method with octree mesh refinement is used. It is implemented in the Gerris (Popinet in J Comput Phys 190(2):572–600, 2003. doi: 10.1016/S0021-9991(03)00298-5 ; J Comput Phys 228(16):5838–5866, 2009. doi: 10.1016/j.jcp.2009.04.042 ) code which allows efficient parallel computations. We measure the scaling properties of the fractal viscous fingering patterns that appear in the numerical simulations. One of these properties is the fractal or Hausdorff dimension $$D_\mathrm{F}$$ . The other is the variation of the area A of the viscous fingering cluster with the length L of its perimeter, which varies as a simple power law $$A \sim L^{\alpha }$$ . The injection of an intermediate-viscosity Newtonian fluid as a second step is also simulated. We are thus able to observe an increase of recovery of the high-viscosity fluid behind the fingering front, due to the reduction of the viscosity contrast. Some of these results are compared to waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sandstone.

There is a gap between direct discrete fracture network numerical model and conceptual multi-linear/porosity analytical model for simulating production performance from fractured unconventional reservoirs. The principle focus of this work is on proposing a hybrid model of complexly fractured reservoir by associating fractal theory with multiple-fractured configuration. The formulation is established on the trilinear-flow idealization presented by Brown et al. (2009). Our model could account for the heterogeneity of fracture network in stimulated reservoir volume (SRV) and the arbitrary properties of multiple hydraulic fractures. Furthermore, a semi-analytical solution is correspondingly presented by incorporating Laplace–Fourier transformation and Stehfest numerical inversion based on the principle of pressure superposition. The new algorithm integrates multiple trilinear-flow solutions for single-fracture hypothesis into a general solution for multi-fractured horizontal well. The second focus is put on verification of the semi-analytical solution by comparing with alternative analytical/numerical simulations for two cases: (a) multi-fractures solution in homogeneity media; (b) multi-fractures solution in heterogeneity media with fractal characteristic. Excellent agreement between alternative simulations and our solutions is achieved. Finally, several synthetic examples are introduced to illustrate the application of semi-analytical solution in the field of pressure transient analysis and discuss the effects of the parameters on transient pressure behavior, including fracture number/spacing, conductivity, fractal characteristic constant and associated anomalous-diffusion constant. The model provides a new knowledge and insight into understanding flow behavior in fractured unconventional reservoirs.

Porous media are interconnected systems, in which the distribution of pore sizes might follow scaling invariant property and will affect the fluid flow through it significantly. Thus, except for a detailed understanding of the fundamental mechanism at pore scale, hard is it to determine the appropriate relationship between the permeability and the basic properties. In this study, in terms of the size distribution and spatial arrangement of the pores, we analytically derived a permeability model using series–parallel flow resistance mode firstly. And then, together with the scaling invariant characteristics of the porosity, specific area and hydraulic tortuosity, the analytical permeability model is reformulated into a fractal permeability–pore structure relationship. The results indicate that: (1) the square of the porosity ( $$\varphi $$ ) is proportional to the permeability in a fractal porous media, not the cubic law described in Kozeny–Carman (KC) equation; (2) the hydraulic tortuosity is a power law model of the minimum particle size with the exponent $$\epsilon (D_{\mathrm{f}}-d)$$ , where $$D_{\mathrm{f}}$$ and d are the pore size fractal dimension and Euclidean space dimension, respectively, while $$\epsilon $$ is a parameter characterizing the spatial arrangement of pores; (3) the KC numerical prefactor is not a constant in fractal porous media. Its value, however, increases linearly with the size ratio of the minimum to the maximum pores but decreases exponentially with $$D_{\mathrm{f}}$$ . More importantly, it is found to be a parameter characterizing the difference of fluid flow in porous media from that in a straight tube described by the Poiseuille law. The performance of the new fractal permeability–pore model is verified by lattice Boltzmann simulations, and the numerical prefactor universality is examined as well.

Classical site percolation is studied on square pore networks generated through the dual site–bond model (DSBM). This model allows different strengths of correlations among the lattice elements in such a way that patches of sites and bonds with similar sizes are structured as correlations are setup. The typical size of these patches depends on the strength of correlations. The aim of this paper is to find whether the global patchy structure is responsible for the patterns found in classical percolation structures. To this end, we measure different static quantities of the correlated networks, such as correlation length, and fractal quantities of the spanning clusters. We observe a significant gap from the reported behavior. For instance, our results indicate that the fractal dimension of the spanning cluster is nonuniversal and changes with the network correlation. The DSBM is good to produce classical percolation clusters with different fractal dimension and different lacunarity behavior. Comparison with other authors’ recent findings is presented.

We investigate a contaminant transport in fractal media with randomly inhomogeneous diffusion barrier. The diffusion barrier is a low-permeable matrix with rare high-permeability pathways (punctures). At times, less than a characteristic matrix diffusion time, the problem is effectively barrier-free, where an effective source acts during the time $$t_\mathrm{eff} \ll t$$ . The punctures result in a precursor contaminant concentration at short times and additional stage of the asymptotic concentration distribution at long times. If the source surface area is large enough, then the barrier can be considered as statistical homogeneous medium; otherwise, strong fluctuations occur.

In this work we present a model for radial flow in highly heterogenous porous media. Heterogeneity is modeled by means of fractal geometry, a heterogeneous medium is regarded as fractal if its Hausdorff dimension is non-integral. Our purpose is to present a derivation of the model consistent with continuum mechanics, capable to describe anomalous diffusion as observed in some naturally fractured reservoirs. Consequently, we introduce fractional mass and a generalized Gauss theorem to obtain a continuity equation in fractal media. A generalized Darcy law for flux completes the model. Then we develop the basic equation for Well test analysis as is applied in petroleum engineering. Finally, the equation is solved by Laplace transform and inverted numerically to illustrate anomalous diffusion. In this case by showing that the flow rate from fractal systems is smaller than that from the Euclidean system.

The mathematical model for transient fluid flow in porous media is based in general on mass conservation principle. Because of the small compressibility of formation fluid, the quadratic term of pressure gradient is always ignored to linearize the non-linear diffusion equation. This may result in significant errors in model prediction, especially at large time scale. In order to solve this problem, it may be necessary to keep the quadratic term in the non-linear equations. In our study, the quadratic term is reserved to fully describe the transient fluid flow. Based on this rigorous treatment, the mathematical models are established to analyze the transient flow behavior in a double porosity, fractal reservoir with spherical and cylindrical matrix. In addition, Laplace transformation method is employed to solve these mathematical models and the type curves are provided to analyze the pressure transient characteristics. This study indicates that the relative errors in calculated pressure caused by ignoring the quadratic term may amount to 10 % in a fractal reservoir with double porosity, which can’t be neglected in general for fractal reservoirs with double porosity at large time scale.

In this study, the conditional moment closure approach, which is proven to be very useful for modelling of reactions in turbulent flows, is extended to characterise adsorbing, desorbing or reacting flows in porous media. A complete specification of the porous distance conditioned moment closure model, which is formulated in terms of single-conditioned expectations, is presented. The closure of the model equations is obtained assuming the diffusion approximation for fluxes of the reactive species. The model simulates complex multi-cascade processes of convective and diffusive transport of species between pores in a continuous and consistent manner and is a generalisation of dual (or triple) porosity concept. The model addresses the major difficulty of describing transport, entrapment and sorption processes in porous media with fractal properties, where distant transport occurs in the largest pores or fractures, while the adsorbing or desorbing surface is mainly allocated in small pores. The model is able to simulate various regimes of methane replacement by CO2 in a coal sample, which makes it useful for optimising the design and parameters of enhanced coal bed methane recovery operations. It is demonstrated that the power-low decrease in downstream methane concentration, which has been observed experimentally, can be accurately reproduced by the model.

By utilizing fractal dimension as one of the parameters to characterize rocks, a mathematical model was derived to predict the production rate by spontaneous imbibition. This fractal production model predicts a power law relationship between spontaneous imbibition rate and time. Fractal dimension can be estimated from the fractal production model using the experimental data of spontaneous imbibition in porous media. The experimental data of recovery in gas-water-rock and oil–water–rock systems were used to test the fractal production model. The rocks (Berea sandstone, chalk, and The Geysers graywacke) in which the spontaneous water imbibition experiments were conducted had different permeabilities ranging from 0.5 to over 1000 md. The results demonstrate that the fractal production model can match the experimental data satisfactorily in the cases studied. The fractal dimension data inferred from the model match were approximately equal to the values of fractal dimension measured using a different technique (mercury-intrusion capillary pressure) in Berea sandstone.

Modeling naturally fractured reservoirs (NFRs) requires an accurate representation of fracture network permeability (FNP). Conventionally, logs, cores, seismic, and pressure transient tests are used as a data base for this. Our previous attempts showed that a strong correlation exists between the fractal parameters of 2-D fracture networks and their permeability (Jafari and Babadagli, SPE 113618, Western Regional and Pacific Section AAPG joint meeting, 2008; Jafari and Babadagli, SPE Reserv Eval Eng 12(3):455–469, 2009a). We also showed that 1-D well (cores-logs) and 3-D reservoir data (well test) may not be sufficient in FNP mapping and that 2-D (outcrop) characteristics are needed (Jafari and Babadagli, SPE 124077, SPE/EAGE reservoir characterization and simulation conference, 2009b). This paper is an extension of those studies, where only 2-D (single-layer, uniform fracture characteristics in z direction) representations were used. In this paper, we considered a more complex and realistic 3-D network system. Two-dimensional random fractures with known fractal and statistical characteristics were distributed in the x- and y directions. A variation of fracture network characteristics in the z direction was presented by a multilayer system representing three different facieses with different fracture properties. Wells were placed in different locations of the model to collect 1-D fracture density and pressure transient data. In addition, five different fractal and statistical properties of the network of each layer were measured. The equivalent FNP was calculated using a commercial software package as the base case. Using available 1-D, 2-D, and 3-D data, multivariable regression analyses were performed to obtain equivalent FNP correlations for many different fracture network realizations. The derived equations were validated against a new set of synthetic fracture networks, and the conditions at which 1-D, 2-D and 3-D data are sufficient to map FNP were determined. The importance of the inclusion of each data type, i.e., 1-D, 2-D and 3-D, in the correlations was discussed. It was shown that using only 3-D data are insufficient to predict the FNP due to wide spatial heterogeneity of the fracture properties in the reservoir, which cannot be captured from single-well tests. Incorporating all types of data (1-D, 2-D, and 3-D) would result in better prediction. Also, it is recommended that the 2-D data of the most conductive layer in reservoir, which has longer fractures with a higher density, should be incorporated in the correlations.

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the \({\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}}\) law is obtained (here DT is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (DT = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.

This work studies the flow characteristics of power-law fluids in the fractal-like tree network. A fractal model is developed for the permeability of power-law fluid flow in fractal-like tree network based on straight capillary model, generalized Darcy’s law and constitutive equation for power-law fluids. Analytical expression for permeability of power-law fluids in the network is presented and found to be a function of network microstructural parameters such as the branching diameter ratio, the branching length ratio, the total number of branching levels, the bifurcation angle, the branching number, the diameter of the zeroth branching level and the power exponent of power-law fluids. Both the phase permeabilities and the relative permeabilities are also derived and found to be a function of power exponent for the wetting phase and non-wetting phase, the saturation and other microstructural parameters and independent of the bifurcation angle.

Motivated by the need to determine the dependencies of two-phase flow in a wide range of applications from carbon dioxide sequestration to enhanced oil recovery, we have developed a standard two-dimensional, pore-level model of immiscible drainage, incorporating viscous and capillary effects. This model has been validated through comparison with several experiments. For a range of stable viscosity ratios (M = μinjected,nwf/μdefending, wf ≥ 1), we had increased the capillary number, Nc and studied the way in which the flows deviate from fractal capillary fingering at a characteristic time and become compact for realistic capillary numbers. This crossover has enabled predictions for the dependence of the flow behavior upon capillary number and viscosity ratio. Our results for the crossover agreed with earlier theoretical predictions, including the universality of the leading power-law indicating its independence of details of the porous medium structure. In this article, we have observed a similar crossover from initial fractal viscous fingering (FVF) to compact flow, for large capillary numbers and unstable viscosity ratios M < 1. In this case, we increased the viscosity ratio from infinitesimal values, and studied the way in which the flows deviate from FVF at a characteristic time and become compact for non-zero viscosity ratios. This crossover has been studied using both our pore-level model and micro-fluidic flow-cell experiments. The same characteristic time, τ = 1/M0.7, satisfactorily describes both the pore-level results for a range of large capillary numbers and the micro-fluidic flow cell results. This crossover should lead to predictions similar to those mentioned above.

The microstructure of a suite of sandstone samples is quantitatively analyzed using a method which combines information from thin section micrographs of the pore space with mercury injection porosimetry in a statistical framework. This method enables the determination of a continuous distribution of pore sizes ranging from few nanometre to several hundred micrometre. The data obtained unify fractal and Euclidean aspects of the void space geometry, yield estimates of the pore-to-throat aspect ratio and challenge the ability of commonly used network models to describe fluid percolation in multiscale porous media. Application of critical path analysis to the prediction of flow permeability and electrical conductivity of sandstone core samples using the new information produces results comparable to those obtained by the classical approach—a fact attributed to the presence of macroscopic heterogeneity at the scale of several millimetres.

Capillary pressure curves of six low porosity and low permeability core samples from The Geysers geothermal field were measured using the mercury-intrusion approach to characterize the heterogeneity of rock. One high permeability Berea sandstone core sample was analyzed similarly, for comparison. The maximum pressure of mercury intruded into the rock was about 200 MPa to reach the extremely small pores. Experimental data showed that the capillary pressure curves of The Geysers rock are very different from that of the Berea sandstone. It was found that the frequently used capillary pressure models could not be used to represent the data from The Geysers rock samples. This might be because of the fractures in the rock. To this end, a fractal technique was proposed to model the features of the capillary pressure curves and to characterize the difference in heterogeneity between The Geysers rock and Berea sandstone. The results demonstrated that the rock from The Geysers geothermal field was fractal over a scaling range of about five orders of magnitude. The values of the fractal dimension of all the core samples (six from The Geysers and one Berea sandstone) calculated using the proposed approach were in the range from 2 to 3. The results showed that The Geysers rock with a high density of fractures had a greater fractal dimension than Berea sandstone which is almost without fractures. This shows that The Geysers rock has greater heterogeneity, as expected.