Appendix C Derivation of Equations for an Air-Water Mixture

Transcription

1 Appendix C Derivtiof Equtions for n Air-Wter Mixture The following ssumptions were mde for this nlysis. We will ssume initil sturtion levels re greter thn 85 percent, which implies tht ll ir bubbles re occluded. Surfce tension effects will be neglected, which llows us to ssert tht the ir bubbles within the wter will be t the sme pressure s the wter. The ir is soluble in wter nd observes Henry's Lw, nd the rte of increse in pore wter pressure from ny simultion is slower thn the rte of diffusiof ir in wter. Finlly, prior to full sturtion, the compressibility or bulk modulus of wter is constnt. We will first develop the equtions for n ir-wter system with rigid porous skeleton, thene with compressible porous skeleton. The following terms re used in the derivtiof the compressibility of n ir-wter mixture. Let V denote the totl volume of ir nd wter, V v the volume of the void spce, the volume of wter, V d the volume of dissolved ir, V the totl volume of ir, V á the volume of free ir, which is equl to V - V d, P w the pore wter pressure, the pore ir pressure nd H Henry's constnt. 250

2 Appendix C Derivtiof Equtions for n Air-Wter Mixture 25 A subscripted "o" is used to indicte n initil vlue. The totl mss of ir nd wter remins constnt. Substituting expressions for porosity (n) nd sturtion (S), the initil volumes of wter o nd free ir o my be expressed s o V vo C. nd o ( ) V vo ( ) C.2 Using Henry's Lw nd Eqution C., the initil volume of dissolved ir my be expressed s V do o H H C.3 The sum of Equtions C.2 nd C.3 yields n expression for the initil totl volume of ir in the system V o ( H ) C.4 Expressions for the compressibility of wter nd n ir-wter mixture my be written s C w d C.5 nd dv d C.6 respectively. Substituting Eqution C.5 into Eqution C.6 one obtins dv C w C.7 We will now use Boyle's Lw to develop n expression for the derivtive in Eqution C.7. Boyle's Lw my be written s

3 Appendix C Derivtiof Equtions for n Air-Wter Mixture 252 V V o o C.8 If we ssume the pore nd ir pressure re equl nd, we cn write the following V d V do o P wo V d P w o V d C.9 from which we write dv o V d V d V o o V 2 2 o C.0 Substituting the ltter expression in Eqution C.0 into Eqution C.7, one obtins V 2 V o o C w C. which is n expression for the compressibility of n ir-wter mixture. By judiciously substituting for the volume terms in Eqution C., we will develop finl expression for the compressibility of the mixture. By combining Boyle's Lw ( Eqution C.8) nd Eqution C.4, we my write n expression for the current totl volume of ir V o ( H ) C.2 The current volume of wter my be expressed s o ( C w /P ) C.3

4 Appendix C Derivtiof Equtions for n Air-Wter Mixture 253 nd, fter substituting for, s o ( C w /P ) C.4 The current volume of dissolved ir, which is functiof Henry's Lw nd the current volume of wter, is written s V d H ( C w /P ) C.5 Subtrcting Eqution C.5 from Eqution C.2, one obtins n expression for the current volume of free ir o ( H ) H ( C w / P ) C.6 Adding Equtions C.6 nd C.4, one obtins o ( H ) ( H )( C w / P ) C.7 which will eventully be substituted bck into Eqution C.. Combining Equtions C.4 nd C.2, one my write V 2 V o o 2 ( H ) C.8 nd, by multiplying Eqution C.4 by the compressibility of wter nd dropping the higher order terms, one obtins C w C w C.9 Substituting Equtions C.7, C.8, nd C.9 into Eqution C. yields the finl expression for the compressibility of n ir-wter mixture

5 Appendix C Derivtiof Equtions for n Air-Wter Mixture 254 o ( H ) ( H )( C w / P ) o 2 ( H ) C w C.20 In similr mnner, n expression for the level of sturtion my be developed nd written s S ( C w / P ) o ( H ) ( H )( C w / P ) C.2 When the porous skeleton is compressible, the current void volume my be expressed s ( 0 kk ) C.22 where is the initil totl volume of voids nd solids nd 0 kk is the effective volumetric strin. Substituting the bove nd Eqution C.4 into the first expression in Eqution C.2, one obtins S ( C w /P ) 0 kk C.23 which is n expression for the level of sturtion in deforming porous skeleton. An eqution for the compressibility of n ir-wter mixture within deforming porous skeleton my be obtined by combining Equtions C., C.8, C.9, nd C.22 to yield

6 Appendix C Derivtiof Equtions for n Air-Wter Mixture kk o 2 ( H ) C w C.24 In the process of clcultion, one must first evlute Eqution C.23. If the level of sturtion is less thne, Eqution C.24 is used to clculted the bulk modulus of the pore fluid. If the level of sturtion is equl to one, the bulk modulus of the pore fluid is clculted from the EOf wter.