William J. Milliken

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.

The effect of surfactant on the transient motion of Newtonian drops

The effect of dilute, insoluble surfactant on the deformation and breakup of a viscous drop is examined. Two cases are considered: the deformation and stretching of a drop in a uniaxial extensional flow and the surface‐tension‐driven motion of an elongated drop in a quiescent fluid. Aside from rescaling the mean capillary force through an average decrease in the interfacial tension, surfactants alter the motion of a viscous drop through gradients in interfacial tension. The effects of surfactants are found to be most pronounced for small viscosity ratios, where Marangoni stresses substantially retard the interfacial velocity and cause the drop to behave as though it were more viscous. Surfactants are found to facilitate the formation of pointed ends during drop stretching, and this may explain the observation of tip streaming in experiments with viscoelastic drops. Surfactant gradients also allow drops to be elongated to a larger degree without producing end pinching.

A New Method for the Scale Up of Displacement Processes in Heterogeneous Reservoirs

A general method for the scale up of highly detailed, heterogeneous reservoir cross sections to coarser scales, for the purposes of flow simulation of displacement processes, is developed and applied. The technique involves the nonuniform coarsening of the fine scale description, with fine 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 reservoir. The method is designed to capture both average behavior as well as some important behaviors which are due to the extremes of the permeability field, such as the breakthrough time of the displacing fluid. The method is applied to several example problems, including two actual field examples, and is shown to provide coarsened models (~ 25 x 25) which give simulation results for fractional flows and saturations in close agreement with fine scale (~ 100 x 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 reservoir description is to a large degree process independent.

Field Experiences with Assisted and Automatic History Matching Using Streamline Models

Reconciling high-resolution geologic models to production history is a very time-consuming process in reservoir modeling. Current practice still involves a tedious historymatching process that is highly subjective and often employs ad-hoc property multipliers. Recently streamline models have shown significant promise in improving the history matching process. In particular, the streamline-based “assisted historymatching” utilizes the streamline trajectories to identify and limit changes only to the regions contributing to the well production history. It is now a well-established procedure and has been applied successfully to numerous field cases. In this paper, we enhance the streamline-based assisted history matching in two important aspects that can significantly improve its efficiency and effectiveness. First, we utilize streamline-derived analytic sensitivities to determine the spatial distribution and magnitude of the changes needed to improve the history match. Second, we use a “generalized travel time inversion (GTTI)” for model updating via an iterative minimization procedure. Using this approach, we can account for the full coupling of the streamlines rather than changing individual or bundles of streamlines at a time. The approach is more akin to automatic history matching. However, by intervening at every step in the iterative model updating, we can retain control over the process as in assisted history matching. Our approach leads to significant savings in time and manpower during field-scale history matching. We demonstrate the power of our method using two field examples with model sizes ranging from 10 5 to 10 6 grid blocks and with over one hundred wells. The reservoir models include faults, aquifer support and several horizontal/high angle wells. History matching is performed using both assisted history matching and the GTTI. Whereas the general trends in permeability changes are similar for both the methods, the GTTI seems to significantly improve the water cut history matching on a well-by-well basis within a few iterations. Our experience indicates that the GTTI can also be used very effectively to improve the quality of history match derived from the assisted history matching. The changes to the reservoir model from GTTI are found reasonable with no artificial discontinuities or apparent loss of geologic realism.

The Effect of Geologic Parameters and Uncertainties on Subsurface Flow: Deepwater Depositional Systems

The application of reservoir simulation as a tool for reservoir development and management is widespread in the oil and gas industry. Moreover, it is recognized that the results of any reservoir simulation model are strongly influenced by the underlying geologic model. However, the direct relationship between geologic parameters and subsurface flow is obscure. In this paper we explore this relationship in a deepwater depositional system using data from two reservoir analogs: the shallow seismic dataset from the Mahakam Fan and outcrop data from the Brushy Canyon Formation of West Texas. Shallow seismic data from the Mahakam Fan area shows a high-resolution deepwater channel-levee system consisting of 10 migrating channels. Using an experimental design framework and a series of three increasingly complex models, we investigated the effect of nine different geologic factors on several different measures of the flow behavior. Our results show that, as expected, different geologic factors influence different measures of the flow. Most significant is the clear effect that the proportion and organization of the different internal facies making up the channels have on the recovery factor and net oil production. The Brushy Canyon outcrops used in this work represent sand-rich proximal deposits of a distributary lobe complex. Here we built models on a very small length scale to investigate the effects of sheet-like reservoir architecture as well as internal facies distribution of the sheets on subsurface flow. Again, an experimental design framework was employed, this time to examine the influence of 11 input variables. The proportion and organization of the internal lobe facies has a significant influence on the subsurface flow here but in these distributary lobe complexes other variables, including the stacking of the lobes, were also found to be important. The models in this study address flow behavior in deepwater, sparse well environments. Using models from the simple to the complex, we found that several parameters incorporated in the complex models, and not in simple models, had a significant impact on the predicted flow.

Apparent viscosity of suspensions of rods using falling ball rheometry

Falling ball rheometry measurements of the apparent relative viscosity of suspensions of randomly oriented rodlike particles are reported for rods having a wide range of aspect ratios and concentrations. These data, combined with those from earlier studies [Milliken et al., J. Fluid Mech. 202, 217 (1989); Powell et al., J. Rheology 33, 1173 (1989)], provide a master curve describing the apparent viscosity in terms of a single parameter, nL3, where L is the length of the particle and n is the number density.

Application of a New Scale Up Methodology to the Simulation of Displacement Processes in Heterogeneous Reservoirs

A general method for the scale up of highly detailed, heterogeneous reservoir cross sections is presented and applied to the simulation of several recovery processes in a variety of geologic settings. The scale up technique proceeds by first identifying portions of the fine scale reservoir description which could potentially lead to high fluid velocities, typically regions of connected, high permeability. These regions are then modeled in detail while the remainder of the domain is coarsened using a general numerical technique for the calculation of effective permeability. The overall scale up method is applied to the cross sectional simulation of three actual fields. Waterflood, steamflood and miscible flood recovery processes are considered. In all these cases, the scale up technique is shown to give coarsened reservoir descriptions which provide simulation results in very good agreement with those of the detailed reservoir descriptions. For these simulations, speedups in computation times, for the coarsened models relative to their fine grid counterparts, range from a factor of 10 to a factor of 200.