Louis J. Durlofsky

NUMERICAL CALCULATION OF EQUIVALENT GRID BLOCK PERMEABILITY TENSORS FOR HETEROGENEOUS POROUS MEDIA

A numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented. The method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield, upon appropriate averaging of the fine scale velocity field, the coarse scale or equivalent grid block permeability. When the region over which this coarse scale permeability is computed constitutes a representative elementary volume (REV), the resulting equivalent permeability may be interpreted as the effective permeability of the region. Solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle-based finite element numerical procedure, which allows for the modeling of geometrically complex features. The specification of periodic boundary conditions is shown to yield symmetric, positive definite equivalent permeability tensors in all cases. The method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields. The applicability and limitations of the method for these more general heterogeneity fields are discussed.

Dynamic simulation of hydrodynamically interacting particles

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

A Coupled Local-Global Upscaling Approach for Simulating Flow in Highly Heterogeneous Formations

A new technique for generating coarse scale models of highly heterogeneous subsurface formations is developed and applied. The method uses generic global coarse scale simulations to determine the boundary conditions for the local calculation of upscaled properties (permeability or transmissibility). An iteration procedure assures consistency between the local and global calculations. Transport processes are simulated using a subgrid velocity reconstruction technique applied in conjunction with the local-global upscaling procedure. For highly heterogeneous (e.g., channelized) systems, the new method is shown to provide considerably more accurate coarse scale models for flow and transport, relative to reference fine scale results, than do existing local (and extended local) upscaling techniques. The applicability of the upscaled models for dierent global boundary conditions is also considered.

Analysis of the Brinkman equation as a model for flow in porous media

The fundamental solution or Greens function for flow in porous media is determined using Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the velocity disturbance caused by a source particle on field particles located throughout a monodisperse porous medium at a given value of volume fraction of solids o, and by considering many such realizations of the (random) porous medium, the fundamental solution is determined. Comparison of this fundamental solution with the Greens function of the Brinkman equation shows that the Brinkman equation accurately describes the flow in porous media for volume fractions below 0.05. For larger volume fractions significant differences between the two exist, indicating that the Brinkman equation has lost detailed predictive value, although it still describes qualitatively the behavior in moderately concentrated porous media. At low o where the Brinkman equation is known to be valid, the agreement between the simulation results and the Brinkman equation demonstrates that the Stokesian dynamics method correctly captures the screening characteristic of porous media. The simulation results for o ≥ 0.05 may be useful as a basis of comparison for future theoretical work.

Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities

The accuracy of the Darcy velocity, flux, and stream function computed from lowest-order, triangle-based, control volume and mixed finite element approximations to the two-dimensional pressure equation is considered. The control volume finite element method, similar to integrated finite difference methods and analogous to the interpolation of Galerkin finite element results over “control volumes,” is shown to yield a conservative velocity field and smooth streamlines. The streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method, which approximates the pressure and velocity variables separately. It is shown that for systems with only moderate degrees of heterogeneity, the control volume finite element method is the more computationally efficient alternative; i.e., it provides more accurate flow results for a given number of unknowns. For more variable or discontinuous permeability fields, by contrast, such as sand/shale systems, the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns.

Experimental study of two and three phase flows in large diameter inclined pipes

Abstract Steady-state and transient experiments of water–gas, oil–water and oil–water–gas multiphase flows were conducted on a transparent 11 m long, 15 cm diameter, inclinable pipe using kerosene, tap water and nitrogen. The pipe inclination was varied from 0° (vertical) to 92° and the flow rates of each phase were varied over wide ranges. Fast-acting valves enabled the trapping of the fluid flowing in the pipe, which in turn allowed for the accurate measurement of the absolute volumetric fraction (holdup) of each phase. A nuclear densitometer, as well as ten electrical probes at various locations along the pipe, provided additional steady-state and transient measurements of the holdup. A total of 444 tests were conducted, including a number of repeated tests. Bubble, churn, elongated-bubble, slug, and stratified/stratified-wavy flows were observed for water–gas and oil–water–gas flows, while dispersed/homogeneous, mixed/semi-mixed and segregated/semi-segregated flows were observed for oil–water flows. Extensive results for holdup as a function of flow rates, flow pattern and pipe inclination are reported, and the various techniques for measuring holdup are compared and discussed. The flow pattern and shut-in holdup are also compared with the predictions of a mechanistic model. Results show close agreement between observed and predicted flow pattern, and a reasonable level of agreement in holdup.

A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media

Abstract A general method for the scale-up of highly detailed, heterogeneous, cross sectional flow models to coarser scales is developed and applied. The technique involves the nonuniform coarsening of the detailed description, with finer resolution introduced in regions of potentially high fluid velocities (typically regions of connected, high permeability) and coarse, homogenized descriptions applied to the remainder of the flow domain. The method is designed to capture both average behavior as well as some important behaviors which are due to the high extremes of the permeability field, such as the breakthrough of the displacing fluid. The method is applied to several example problems, including two actual oil field examples, and is shown to provide coarsened models (∼ 25 × 25) which give simulation results for fractional flows and saturations in close agreement with fine scale (∼ 100 × 100) results. These examples demonstrate the ability of the method to capture a wide variety of flow behavior without the need for specific knowledge of the global flow field, indicating that the coarsened description of the formation is, to a large degree, process independent.

Implementation of Adjoint Solution for Optimal Control of Smart Wells

Practical production optimization problems typically involve large, highly complex reservoir models, thousands of unknowns and many nonlinear constraints, which makes the numerical calculation of gradients for the optimization process impractical. This work explores a new algorithm for production optimization using optimal control theory. The approach is to use the underlying simulator as the forward model and its adjoint for the calculation of gradients. Direct coding of the adjoint model is, however, complex and time consuming, and the code is dependent on the forward model in the sense that it must be updated whenever the forward model is modified. We investigate an adjoint procedure that avoids these limitations. For a fully implicit forward model and specific forms of the cost function and nonlinear constraints, all information necessary for the adjoint run is calculated and stored during the forward run itself. The adjoint run then requires only the appropriate assembling of this information to calculate the gradients. This makes the adjoint code essentially independent of the forward model and also leads to enhanced efficiency, as no calculations are repeated. Further, we present an efficient approach for handling nonlinear constraints that also allows us to readily apply commercial constrained optimization packages. The forward model used in this work is the General Purpose Research Simulator (GPRS), a highly flexible compositional/black oil research simulator developed at Stanford University. Through two examples, we demonstrate that the linkage proposed here provides a practical strategy for optimal control within a general-purpose reservoir simulator. These examples illustrate production optimization with conventional wells as well as with smart wells, in which well segments can be controlled individually.

Computing permeability of fault zones in eolian sandstone from outcrop measurements

The large-scale equivalent permeabilities of strike-slip faults in porous sandstone are computed from detailed field measurements. The faults, which occur in the Valley of Fire State Park, Nevada, were previously characterized, and the flow properties of their individual features were estimated. The faults formed from the shearing of joint zones and are composed of a core of fine-grain fault rock (gouge) and deformation bands and a peripheral damage zone of joints and sheared joints. High-resolution fault-zone maps and permeability data, estimated using image analysis calibrated to actual measurements, are incorporated into detailed finite difference numerical calculations to determine the permeability of regions of the fault zone. Faults with slips of magnitude 6, 14, and 150 m are considered. The computed fault-zone permeabilities are strongly anisotropic in all cases. Permeability enhancement of nearly 1 order of magnitude (relative to the host rock) is observed for the fault-parallel component in some regions. Fault-normal permeability, by contrast, may be 2 orders of magnitude less than the host rock permeability. The fault-normal permeability is a minimum for the fault with the highest slip. For a representative fault region, the fault-parallel component of permeability is highly sensitive to the fracture aperture, although the fault-normal permeability is insensitive. The procedures developed and applied in this article can be used for any type of fault for which detailed structural and permeability data are available or can be estimated.

Use of Border Regions for Improved Permeability Upscaling

A procedure for the improved calculation of upscaled grid block permeability tensors on Cartesian grids is described and applied. The method entails the use of a border region of fine-scale cells surrounding the coarse block for which the upscaled permeability is to be computed. The implementation allows for the use of full-tensor permeability fields on the fine and coarse scales. Either periodic or pressure–no flow boundary conditions are imposed over the extended local domain (target block plus border regions) though averaged quantities, used to compute the upscaled permeability tensor, are computed only over the target block region. Flow and transport results using this procedure are compared to those from standard methods for different types of geological and simulation models. Improvement using the new approach is consistently observed for the cases considered, though the degree of improvement varies for different models and flow quantities.