Linear Transient Flow Solution for Primary Oil Recovery with Infill and Conversion to Water Injection
Authors: Zwahlen Eric D. , Patzek Tad W. 
Published: December 2007

Abstract
We present analytical solutions for primary production, producer infills and early response to waterflood in a low permeability, compressible, layered reservoir filled with oil, water and gas. The sample calculations are for the California Diatomites, but the equations apply to other tight rock systems. Primary oil recovery from rows of hydrofractured wells in a low permeability reservoir is described by linear transient flow of oil, water and gas with the concomitant pressure decline. During primary, it may be desirable to drill infill wells to accelerate oil production. At some later time, the infill wells may be converted into waterflood injectors for pressure support and incremental oil recovery. In this paper, we analyze the pressure response and fluid flow rates due to the original wells and infill wells drilled halfway between the original wells, and - finally - due to water injection at the infill wells. All of the formation and fluid properties are described by a single hydraulic diffusivity, a, assumed to be independent of time and production or injection. We solve the one-dimensional pressure diffusion equation analytically using pressure boundary conditions at the original and infill wells and use superposition to account for the water injection. We give solutions for the pressure in the formation and discuss how to calculate oil, water and gas rates and cumulatives as functions of time at both the original wells and infill wells. Finally, we present a computational example of oil production from a stack of seven diatomite layers with different properties and show the effects of infill wells and water injection on the total oil production. We show that results of this analytical solution and a compositional numerical simulation for primary production in the diatomite agree well. Our analysis can predict the onset of pressure depletion and quantify how long to produce from the infill wells before injecting water. It shows that producing from the infill well for a few years significantly increases the production from the field and can minimize the lost production at the infill well due to conversion to a waterflood injector. Our analysis also generates very reliable, well-by-well, field-wise forecasts of fluid production and water injection.

Introduction
The late and middle Miocene diatomaceous oil fields in the San Joaquin Valley, California, are located in Kern County, some forty miles west of Bakersfield. An estimated original-oil-in-place in the Monterey diatomaceous fields exceeds 10 billion barrels and is comparable to that in Prudoe Bay in Alaska.
Cyclic bedding of the diatomite is a well documented phenomenon, attributed to alternating deposition of detritus beds, clay, and biogenic beds. The cycles span length-scales that range from a fraction of an inch to tens of feet, reflecting the duration of depositional phases from semi-annual to thousands of years. On a large scale, there are at least seven distinct oil producing layers with good lateral continuity within each layer, but little vertical continuity between adjacent layers. The diatomaceous rocks are very porous (25–65 percent), rich in oil (35–70 percent), and nearly impermeable (0.1–10 millidarcy). The high porosity and oil saturation, together with large thickness (up to 1000 feet) and area (up to a few square miles per field) translate into the gigantic oil-in- place estimates.
To compensate for the low reservoir permeability, all wells in the diatomite must be hydrofractured. A typical well has 3–8 fractures with an average fracture half-length of 150 feet. Wells are usually spaced along lines following the maximum in-situ stress every 330 feet (2–1/2 acre), 165 feet (1–1/4 acre) or even 82 feet (5/8 acre). Thousands of hydrofractures have been already induced and thousands more may be created as new recovery processes, such as waterflood or steam drive on 5/8 acre spacing, become commercially viable.
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