The use of renormalization for calculating effective permeability

There is a need in the numerical simulation of reservoir performance to use average permeability values for the grid blocks. The permeability distributions to be averaged over are based on samples taken from cores and from logs using correlations between permeabilities and porosities and from other sources. It is necessary to use a suitable effective value determined from this sample. The effective value is a single value for an equivalent homogeneous block. Conventionally, this effective value has been determined from a simple estimate such as the geometric mean or a detailed numerical solution of the single phase flow equation.If the permeability fluctuations are small then perturbation theory or effective medium theory (EMT) give reliable estimates of the effective permeability. However, for systems with a more severe permeability variation or for those with a finite fraction of nonreservoir rock all the simple estimates are invalid as well as EMT and perturbation theory.This paper describes a real-space renormalization technique which leads to better estimates than the simpler methods and is able to resolve details on a much finer scale than conventional numerical solution. Conventional simulation here refers to finite difference (or element) techniques for solving the single phase pressure equation. This requires the pressure and permeability at every grid point to be stored. Hence, these methods are limited in their resolution by the amount of data that can be stored in core. Although virtual memory techniques may be used they increase computer time. The renormalization method involves averaging over small regions of the reservoir first to form a new averaged permeability distribution with a lower variance than the original. This pre-averaging may be repeated until a stable estimate is found. Examples are given to show that this is in excellent agreement with computationally more expensive numerical solution but significantly different from simple estimates such as the geometric mean.     

Renormalization calculations of immiscible flow

Oil reservoir properties can vary over a wide range of length scales. Reservoir simulation of the fluid flow uses numerical grid blocks have typical lengths of hundreds of metres. We need to specify meaningful values to put into reservoir engineering calculations given the large number of heterogeneities that they have to encompass. This process of rescaling data results in the calculation of effective or pseudo rock properties. That is a property for use on the large scale incorporating the many heterogeneities measured on smaller scales.For single phase flow, a variety of techniques have been tried in the past. These range from very simple statistical estimates to detailed numerical simulation. Unfortunately, the simple estimates tend to be inaccurate in real applications and the numerical simulation can be computationally expensive if not impossible for very fine grid representations of the reservoir. Likewise, pseudorelative permeabilities are time consuming to generate and often inaccurate.Real-space renormalization is an alternative technique which has been found to be computationally efficient and accurate when applied to single-phase flow. This approach solves the problem regionally rather than trying to solve the whole problem in one simulation. The effective properties of small regions are first calculated and then placed on a coarse grid. The grid is further coarsened and the process repeated until a single effective property has been calculated. This has enabled calculation of effective permeability of extremely large grids to be performed, up to 540 million grid blocks in one application.This paper extends the renormalization technique to two-phase fluid flow and shows that the method is at least 100 times faster than conventional pseudoization techniques. We compare the results with high resolution numerical simulation and conventional pseudoization methods for three different permeability models. We show that renormalization is as accurate as the conventional methods when used to predict oil recovery from heterogeneous systems.     

Effective properties for flow calculations

In this paper we discuss the background to the problems of finding effective flow properties when moving from a detailed representation of reservoir geology to a coarse gridded model required for reservoir performance simulation. In so doing we synthesize the pictures of permeability and transmissibility and show how they may be used to capture the effects of the boundary conditions on the upscaling. These same concepts are applied to the renormalization method of calculating permeability, to show its promise as an accurate, yet fast method.     

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

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

Vertical Permeability Distribution of Reservoirs with Impermeable Barriers

The presence of impermeable horizontal barriers (such as shales) in a reservoir is known to have a significant effect on its vertical permeability. Since calculation of an effective vertical permeability of such a reservoir is important, approximation of the distribution of vertical permeability may also be useful for analysis of the two-phase vertical flow of buoyant fluid, such as may occur in the subsurface injection of carbon dioxide into saline formations. In this situation, the maximum likely vertical permeability of a reservoir with impermeable barriers, which could be estimated from the probability distribution of the vertical permeability, is a more useful metric than an overall effective value for the vertical permeability due to its presumed relationship to breakthrough time. In this article, we derive expressions for the mean and variance of the vertical permeability of both two and three-dimensional reservoirs using the statistical streamline method of Begg and King (Paper No. 13529, 1985), and calculate the probability distribution of the vertical permeability of a reservoir with impermeable barriers. In addition, we also provide a simple statistical analysis of the presence of high vertical permeability regions in the reservoir, which may be of importance in coarse-scale simulations of vertical migration.     

Lattice Boltzmann Simulation of Fluid Flow in Synthetic Fractures

Fractures play an important role in reservoir engineering as they dominate the fluid flow in the reservoir. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. As experimental methods are expensive and time consuming most of the time numerical methods are used to study flow and transport in a fractured rock. In this work, we present results of the numerical computations for single phase flow simulations through two-dimensional synthetically created fracture apertures. These synthetic rock fractures are created using different fractal dimensions, anisotropy factors, and mismatch lengths. Lattice Boltzmann method (LBM), which is a new computational approach suitable to simulate fluid flow especially in complex geometries, was then used to determine the permeability for different fractures. Regions of high velocity and low velocity flow were identified. The resulting permeability values were less than the ones obtained with the cubic law estimates. It has been found that as the mean aperture–fractal dimension ratio increased permeability increased. Moreover as the anisotropy factor increased permeability decreased. Neural network simulations were used to generalize the results.     

Heat loss calculation in thermal simulation

An appraisal is presented for four different methods that are usually incorporated in thermal simulators to estimate the rate of heat loss to surroundings. The methods are the analytical solution using a superposition theorem, the analytical solution using a numerical approximation to the convolution integral, the semi-analytical solution, and the numerical solution. This appraisal includes expressing the equations in a form that can be incorporated into a fully implicit simulator, computer programming complexity, and the computer CPU time and memory storage requirements. A steam flood problem is used for the comparison, and the gas recovery, oil recovery, and heat loss performances for a reservoir in one and two dimensions are presented. It is found that the numerical solution is sensitive to grid size in the overburden, the semi-analytical solution is the simplest to program but its prediction is the least accurate, the analytical solution is the most expensive, whereas the analytical-numerical solution combines both accuracy and acceptable storage requirements, and therefore, it is recommended for use in thermal simulation.     

Experimental study of filmwise condensation considering wave initiation

The problem of wavy filmwise condensation was recently investigated using both analytical and numerical concepts. The analytical solution assumed the film profile as a simple sinwave moving with a constant phase velocity and the problem was reduced to a closed integral form. While the numerical solution considers variations in the wave length and amplitude of the proposed sin wavy free surface. In order to find out the validity of the suggested theoretical models, an experimental test rig was installed, where the experimental parameters were controlled to generate data in the range of 40 <Re < 200. The results of this work demonstrate the possibility of accurate simulation of long vertical condenser using laboratory set of a reasonable dimensions. This simulation yields better results when the thermal and hydrodynamical average quantities at the entrance are properly corrected. Good agreement was found by comparing the generated experimental data, together with others data, with the analytical and numerical models. Imperical formulas for heat transfer were considered and differences between the correlation coefficients was found to be mainly due to unproper definition of the validity ranges of the used dimensionless groups. More realistic definition for the range of validity was introduced with these imperical formulas.     

Upscaling permeability: Error analysis for renormalization

In this paper we briefly discuss the background to the problems of finding effective flow properties when moving from a detailed representation of reservoir geology to a coarse gridded model required for reservoir performance simulation. The basic requirements for the upscaled properties are also discussed. We then consider one technique, renormalization, that in recent years has shown promise as an accurate, yet fast, method. The mathematical background of the renormalization approach is examined. A rigorous formalism is developed that allows an explicit calculation of the error terms to be made. In a very simple case use of the correction terms is shown to produce a dramatic improvement in accuracy of the method.     

An Alternative Approach to Predicting Reservoir Performance in a Coalbed Methane (CBM) Well Flowing Under Dominant Matrix Shrinkage Effect

Coalbed methane (CBM) reservoirs contain gas molecules in adsorbed state into the solid matrix of coal. The pressure depletion in CBM reservoir causes the matrix gas to desorb into the cleat system which leads to matrix shrinkage. The pore volume of the cleat network changes as coal matrix shrinks. Consequently, cleat porosity and permeability of reservoir change as reservoir pressure depletes. The change in cleat porosity and permeability due to shrinkage of coal matrix with depletion of reservoir pressure invalidates the underlying assumptions made in the derivation of diffusivity equation. Under the conditions of changing porosity and permeability, the utility of the standard method of inflow performance relationship (IPR), paired with \(\frac{P}{Z^{*}}\) method suggested by King (in: SPE Annual Technical Conference and Exhibition, New Orleans, 1990), for performance prediction diminishes. In this paper, an effort has been made to predict reservoir performance of such CBM reservoirs with an alternative approach. The method suggested by Upadhyay and Laik (Transp Porous Media, 2017. doi: 10.1007/s11242-016-0816-6) has been leveraged to describe pseudo-steady-state flow in the form of a new equation that relates stress-dependent pseudo-pressure function with time. The analytical equation derived in this paper is useful in predicting reservoir pressure and flowing bottom hole pressure of a CBM well under the situation when coal matrix shrinks below desorption pressure. The paper aims to predict production performance of CBM reservoirs producing under the influence of matrix shrinkage effect with an approach alternative to conventional IPR approach paired with \(\frac{P}{Z^{*}}\) method. The results of this analytical solution have been validated with the help of numerical simulator CMG–GEM as well as in-field production data. The equations and workflow suggested in this paper can be easily implemented in spreadsheet applications like Microsoft Excel tools.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

Numerical Simulation of Single-Phase and Multiphase Non-Darcy Flow in Porous and Fractured Reservoirs

A numerical method as well as a theoretical study of non-Darcy fluid flow through porous and fractured reservoirs is described. The non-Darcy behavior is handled in a three-dimensional, multiphase flow reservoir simulator, while the model formulation incorporates the Forchheimer equation for describing single-phase or multiphase non-Darcy flow and displacement. The non-Darcy flow through a fractured reservoir is handled using a general dual-continuum approach. The numerical scheme has been verified by comparing its results against those of analytical methods. Numerical solutions are used to obtain some insight into the physics of non-Darcy flow and displacement in reservoirs. In addition, several type curves are provided for well-test analyses of non-Darcy flow to demonstrate a methodology for modeling this type of flow in porous and fractured rocks, including flow in petroleum and geothermal reservoirs.     

A Simple Model of a Two-Layered High-Temperature Liquid-Dominated Geothermal Reservoir as a Part of a Large-Scale Hydrothermal Convection System

The Kakkonda geothermal reservoir, Japan, is a typical high-temperature liquid-dominated geothermal reservoir, except for its distinctive two-layered temperature structure. It has a shallow permeable reservoir of 230–260°, and a deep less permeable reservoir of 350–360°. Geology and hydrology indicate that the shallow reservoir is one to two orders of magnitude more permeable than the deep reservoir, but that the two reservoirs communicate. It has been widely assumed in engineering and scientific circles that the connection between the two reservoirs is a zero or low permeability barrier to fluid flow. We show that this hypothesis is untenable, based on both physical evidence and numerical simulation. We numerically model the evolution of the geothermal system as it heats after emplacement of an intrusion. The two-layered temperature structure is found to be a consequence of the permeability difference, i.e. the two-layered permeability structure.     

An Explicit Physics-Based Model for the Transverse Permeability of Multi-Material Dual Porosity Fibrous Media

A simple, explicit model for the transverse permeability of porous media comprised of alternating layers of tows of varying permeability has been proposed. The development of the model is based on results of numerical simulations as well as an earlier developed analytical model for the permeability of such systems (Markicevic and Papathanasiou, 2003). Based on the above, as well as on physical arguments regarding the flow across such dual porosity media, we formulate an explicit model in terms of dimensionless permeabilities. Extensive validation of the model is carried out through numerical simulations using the CFD package FIDAP. A good agreement between the permeabilities predicted by the simple explicit model and those calculated numerically is found. From this model, the form of a master curve for the permeability of such system has been deduced.     

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

A family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2 :259–290; Comput. Geo. 2002; 6 :433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51 :1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve:

• (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice.
• (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution.
• (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution.

The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of Vertical Heterogeneity on Long-Term Migration of CO<Subscript>2</Subscript> in Saline Formations

For deep injection of CO₂ in thick saline formations, the movements of both the free gas phase and dissolved CO₂ are sensitive to variations in vertical permeability. A simple model for vertical heterogeneity was studied, consisting of a random distribution of horizontal impermeable barriers with a given overall volume fraction and distribution of lengths. Analytical results were obtained for the distribution of values for the permeability, and compared to numerical simulations of deep CO₂ injection and convection in heterogeneous formations, using multiple realizations for the permeability distribution. It is shown that for a formation of thickness H, the breakthrough times in two dimensions for deep injection scale as H ² for moderate injection rates. In comparison to heterogeneous shale distributions, a homogeneous medium with equivalent effective vertical permeability has a longer breakthrough time for deep injection, and a longer onset time for convection.     

Predicting Vertical Flow Barriers Using Tracer Diffusion in Partially Saturated,Layered Porous Media

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in \(^{87}\hbox {Sr}/^{86}\) Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.     

Homogenization of Hydraulic Conductivity for Hierarchical Sedimentary Deposits at Multiple Scales

Based on a three-dimensional heterogeneous aquifer model exhibiting non-stationary, statistically anisotropic correlation, three hydrostratigraphic models (HSMs) are created within a sedimentary hierarchy. A geostatistical analysis of natural log conductivity (lnK) is conducted for the units of the HSMs. Hydraulic conductivity is then upscaled using numerical and analytical methods. Increasing lnK variances are evaluated. Results suggest that for the aquifer model tested: (1) the numerical method is capable of upscaling irregular domains with reasonable accuracy for a lnK variance up to 7.0. (2) Accuracy of the upscaled equivalent conductivities (K*) and associated performance of the HSMs are sensitive to homogenization level, heterogeneity variance, and boundary condition. Variance is found to be the most significant factor impacting the accuracy of the HSMs. (3) Diagonal tensor appears a good approximation for the full-tensor K*. (4) For the HSM units, when the variance is low (less than 1.0), all analytical methods are nearly equally accurate; however, when variance becomes higher, analytical methods generally are less accurate.     

Asymptotics,Saturation Fronts,and High Resolution Reservoir Characterization

Using an asymptotic methodology we formulate a fast, accurate algorithm for the inversion of multi-phase flow data. The approach is appropriate for many common reservoir production strategies such as CO₂ and water flooding. The technique compares well to a purely numerical method with a significant reduction in computation time. In an application to fractional flow data from the North Robertson field in West Texas, 100,000 permeability and porosity parameters are determined on a workstation. Generally, higher permeability, approaching 1milli-Darcy, is found in the eastern portion of the reservoir. The permeability estimates agree with type curve analysis for material and volumetric balances and a previous numerical pilot-point inversion.     

A New Cleat Volume Compressibility Determination Method and Corresponding Modification to Coal Permeability Model

Coal is known as a dual-porosity media composed of cleat and matrix pore. Methane can be stored in the cleats or adsorbed on the inner surface of matrix pore. While fluid mobility is mainly controlled by the developed cleat network, methane desorption has a significant effect on cleat deformation. In the process of coalbed methane recovery, both reservoir compaction and matrix shrinkage will occur and have opposite effects on permeability evolutions. A variety of analytical permeability models have been developed to describe the transient characteristics of permeability in coals. In this study, three common permeability models are first revisited and evaluated against the experimental data under uniaxial strain condition. Shi–Durucan (S&D) model demonstrates the best performance among these models. However, constant cleat volume compressibility was used to assume for S&D model, and the generalization of S&D model is significantly limited. For ease of generalization, the relation between cleat volume compressibility and effective horizontal stress is re-derived and introduced to the derivation of permeability model. Since coal reservoirs usually demonstrate strong anisotropy and heterogeneity, the influences of elastic and adsorption properties are further tested to reveal the overall trend of permeability. The results show that S&D model and its modification with the main variable of effective horizontal stress have the best performances in matching the experimental data under uniaxial strain. The relationship between cleat volume compressibility and effective horizontal stress can be better reflected by the inverse proportional function. In addition, the strengths of reservoir compaction effect relative to matrix shrinkage effect in different models only vary with Poisson’s ratio, while their magnitudes are also affected by Young’s modulus. For a typical coal reservoir, the C&B and P&M models will observe a stronger permeability decline at the initial, while the improved P&M model will receive an earlier and more rapid rebound than the S&D and W&Z models.