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Matrix Stress EquationZifeng Li

Yanshan University, China

Abst ract

-1

f ob'z

p p = (5)In porous medium, matrix and fluids bear forces together. Foranalyzing strength and deformation in porous medium, matrixstresses must be calculated. After Terzaghi advanced matrixstress for saturated soil in 1925, many people have used matrix

Structural Matrix Stressstress in their researches. It is found that there is mistake inTerzaghis equation. A new matrix stress equation is established.

If total structural stresses are expressed as

Introduction

In porous medium, matrix and fluids bear forces together. Foranalyzing strength and deformation in porous medium, matrixstresses must be calculated. After Terzaghi advanced matrixstress (effective stress) for saturated soil 1 in 1925, many peoplehave taken part in research of effective stress for porousmedium. In petroleum engineering textbooks 1-4 and monographs,Terzaghi equation is still in use.

It is found that there is mistake in Terzaghis equation. A newmatrix stress equation is established.

Matrix Stress EquationIt is defined in textbooks and monographs that overburden

pressure at some point equals the weight of matrix and fluid in porous medium:

])1[( f mob += gh p (1)or

hg ph

d ])1[(0 f mob += (2)

where, =overburden pressure;ob p g = acceleration of gravity;

= depth of point from surface;h = porosity; m = densityof matrix; f = density of fluid.

Matrix and fluids bear overburden pressure together. The portion that is not acting on fluids is called matrix stress.

Matrix stress advanced by Terzaghi for saturated soil is 1

f ob'z p p = (3)where, = matrix stress in vertical direction; =formation pressure.

'z f p

Terzaghis equation does not consider the effect of porosity. Itis not right theoretically.

Because matrix and fluids bear overburden pressure together,thus

f 'zob )1( p p += (4)

and

obzyx ,, p= , then matrix stresses are'z

'y

'x ,,

=

=

=

1

1

1

f z'z

f y'y

f x'x

p

p

p

(6)

In formation fracturing prediction and oil enhanced fracturingetc, matrix stress (3) is used. These technologies should berebuilt, such as research in reference 5.

ConclusionTerzaghis matrix stress equation (3) is wrong. True matrix

stress equation is (5).

References

1. Huang Wenxi, Engineering Properties of Soil; pp. 248, Waterand Electric Power Press, Beijing, 1983. (In Chinese)

2. Thomas O. Allen, Alan P. Roberts, Production Operations-Well Completion, Workover, and Stimulation(Volume 2); pp.144, Oil & Gas Consultants International, Inc., Tulsa, OK,1978.

3. H.C.H. Darly, George R. Gray, Composition and Properties ofDrilling and Completion Fluids (Fifth Edition); pp. 435, GulfPublishing Company, Houston, TX, 1991.

4. Chen Taoping, Hu Jingbang, Petroleum Engineering; pp.162-172, Petroleum Industry Press, Beijing, 2000. (InChinese)

5. Li Jingyuan, Li Zifeng, Rock Elastic-plastic Stress around aWellbore and Its Stability under Permeation Osmosis;

Engineering Mechanics, vol. 14, No.1, pp.131-137, January1997. (In Chinese)

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Au thor s Biograph ies

Zifeng Li is a professor of YanshanUniversity, China, Technical Editor ofSPE Drilling & Completion. Before

joining Yanshan University, he was first a professor of Daqing Petroleum Instituteand then a professor of China Universityof Geosciences. He has published 80

papers and 4 books on drill string, casing,rod pumping mechanics & well borestability. He holds a BS degree in drilling

engineering and an MS degree in machinery engineering fromDaqing Petroleum Institute and a PhD degree in petroleumdevelopment engineering from Petroleum University, China.